Zhiguo Zeng

Rui Kang

Meilin Wen

Enrico Zio

A Model-Based Reliability Metric Considering Aleatory and Epistemic Uncertainty

Keywords: Reliability, physics-of-failure, epistemic uncertainty, model uncertainty, belief reliability

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I. INTRODUCTION

Reliability refers to the ability of a component or system to perform a required function for a given period of time when used under stated operating conditions [START_REF] Ebeling | An introduction to reliability and maintainability engineering[END_REF]. Traditionally, reliability is measured by the probability that functional failure does not occur in the considered period of time and failure data are used for its estimation based on statistical methods [START_REF] Meeker | Statistical methods for reliability data[END_REF]. In practice, however, failure data are often scarce (if available at all), which defies the use of classical statistical methods and challenges Bayesian methods with respect to the assumption of subjective prior distributions [START_REF] Aven | Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-probabilistic Methods[END_REF]. Due to the problem of limited failure data, model-based methods (cf. physics-of-failure (PoF) methods [START_REF] Pecht | The reliability physics approach to failure prediction modelling[END_REF], structural reliability methods [START_REF] Collins | Mechanical design of machine elements and machines[END_REF], etc.) are widely applied to predict reliability, by deterministically describing the degradation and failure processes using deterministic failure behavior models. More specifically, it is assumed that:

1) the failure behavior of a component or a system can be described by a deterministic model; 2) random variations in the variables of the deterministic model are the sole source of uncertainty. The probabilistic quantification of reliability is, then, obtained by propagating uncertainties through the model analytically or numerically, e.g. by Monte Carlo simulation [START_REF] Mohaghegh | A probabilistic physics-of-failure approach to common cause failures in reliability assessment of structures and components[END_REF][START_REF] Zio | The Monte Carlo simulation method for system reliability and risk analysis[END_REF][START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF].

The random variations represent the uncertainty inherent in the physical behavior of the system and are referred to as * Email of the corresponding author: kangrui@buaa.edu.cn aleatory uncertainty [START_REF] Aven | A new perspective on how to understand, assess and manage risk and the unforeseen[END_REF]. However, the model-based methods are also subject to epistemic uncertainty due to incomplete knowledge on the degradation and failure processes [START_REF] Kiureghian | Aleatory or epistemic? does it matter?[END_REF][START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF]. According to Aven and Zio [START_REF] Aven | Model output uncertainty in risk assessment[END_REF][START_REF] Bjerga | An illustration of the use of an approach for treating model uncertainties in risk assessment[END_REF], epistemic uncertainty may arise because:

1) the deterministic model cannot exactly describe the failure process, e.g., due to incomplete understanding of the failure causes and mechanisms (model uncertainty, also known as structural uncertainty); 2) the precise values of the model parameters might not be accurately estimated due to lack of data in the actual operational and environmental conditions (parameter uncertainty).

In this paper, we introduce a new reliability metric, belief reliability, to explicitly consider the effect of epistemic uncertainty on the model-based methods. For illustrative purposes, we consider only model uncertainty in this paper. However, the framework can be easily extended to deal with parameter uncertainty.

In literature, various approaches have been developed to consider model uncertainty. Mosleh and Droguett reviewed a number of approaches for model uncertainty assessment and compared them in terms of theoretical foundations and domains of application [START_REF] Mosleh | Model Uncertainty: Its Characterization and Quantification[END_REF][START_REF] Droguett | Bayesian treatment of model uncertainty for partially applicable models[END_REF]. Among them, the alternate hypotheses approach and the adjustment factor approach are two most widely applied ones [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF]. The alternate hypotheses approach identifies a family of possible alternate models and probabilistically combines the predictions of them based on Bayesian model averaging, where the probability of each model is evaluated from experimental data or expert judgements [START_REF] Draper | Assessment and propagation of model uncertainty[END_REF][START_REF] Droguett | Bayesian methodology for model uncertainty using model performance data[END_REF]. Apostolakis [START_REF] Apostolakis | The concept of probability in safety assessments of technological systems[END_REF] addressed the issue of model uncertainty in probabilistic risk assessment using the alternate hypotheses approach. Park and Grandhi [START_REF] Park | Quantifying multiple types of uncertainty in physics-based simulation using bayesian model averaging[END_REF] quantified the model probability in the alternate hypotheses approach by the measured deviations between experimental data and model predictions. In [START_REF] Zhang | Model uncertainty and bayesian updating in reliability-based inspection[END_REF], two crack models were probabilistically combined using the alternate hypotheses approach to estimate the failure probability of a butt weld. Other applications of the alternate hypotheses approach include sediment transport models [START_REF] Sabatine | Evaluation of parameter and model uncertainty in simple applications of a 1d sediment transport model[END_REF], identification of benchmark doses [START_REF] Kim | Estimation of a benchmark dose in the presence or absence of hormesis using posterior averaging[END_REF], precipitation modeling [START_REF] Dyrrdal | Bayesian hierarchical modeling of extreme hourly precipitation in norway[END_REF], etc.

In the adjustment factor approach, the model uncertainty is addressed by modifying a benchmark model (the one that we have highest confidence in) with an adjustment factor, which is assumed to be uncertain, and is either added to or multiplied by the prediction results of the model [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF][START_REF] Riley | Quantification of modelform and predictive uncertainty for multi-physics simulation[END_REF]. In [START_REF] Mosleh | The assessment of probability distributions from expert opinions with an application to seismic fragility curves[END_REF], the adjustment factor approach was used to combine experts' estimates according to Bayes' theorem. Zio and Apostolakis [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF] used the approach to assess the risk of radioactive waste repositories. Fischer and Grandhi [START_REF] Fischer | Utilizing an adjustment factor to scale between multiple fidelities within a design process: A stepping stone to dialable fidelity design[END_REF] applied an adjustment factor to low-fidelities models so as to scale them to high-fidelity models. In a series of studies conducted by Park and Grandhi [START_REF] Riley | Quantification of modelform and predictive uncertainty for multi-physics simulation[END_REF][START_REF] Park | A bayesian statistical method for quantifying model form uncertainty and two model combination methods[END_REF][START_REF] Riley | Quantification of modeling uncertainty in aeroelastic analyses[END_REF][START_REF] Park | A bayesian approach for quantification of model uncertainty[END_REF], the adjustment factor approach was combined with the alternate hypotheses approach by introducing an adjustment factor to quantify the uncertainty in each alternate model; the model uncertainty was, then, evaluated by averaging all the models according to the alternate hypotheses approach.

The alternate hypotheses approach requires enumerating a set of mutually exclusive and collectively exhaustive models [START_REF] Droguett | Bayesian treatment of model uncertainty for partially applicable models[END_REF]. In the case of model-based reliability methods, however, it is impossible for us to enumerate all the possible models, which limits the application of the alternate hypotheses approach. Hence, we adopt the adjustment factor approach in this paper to develop a new reliability metric to describe the effect of epistemic uncertainty (model uncertainty) on the modelbased reliability methods.

In the adjustment factor approaches, epistemic uncertainty is quantified by the adjustment factor, which is often determined based on validation test data (for example, see [START_REF] Droguett | Bayesian methodology for model uncertainty using model performance data[END_REF] or [START_REF] Park | A bayesian approach for quantification of model uncertainty[END_REF]). In practice, however, due to limited time and resources, it is hard, if not impossible, to gather sufficient validation test data. Resorting to expert judgements might offer an alternative solution (for example, see [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF]), but they could be criticized for being too subjective. On the other hand, epistemic uncertainty relates to the knowledge on the component or system functions and failure behaviors: as this knowledge is accumulated, epistemic uncertainty is reduced. In the life cycle of a component or system, the knowledge is gained by implementing a number of reliability analysis-related engineering activities, whose purpose is to help designers better understand potential failure modes and mechanisms. For example, through Failure Mode, Effect and Criticality Analysis (FMECA), potential failure modes and their effects could be identified, so that the designer can better understand the product's failure behaviors [START_REF] Carlson | Effective FMEAs: Achieving safe, reliable, and economical products and processes using failure mode and effects analysis[END_REF]. Similar engineering activities include Failure Report, Analysis, and Corrective Action System (FRACAS) [START_REF] Benbow | The Certified Reliability Engineer Handbook[END_REF], Reliability Growth Test (RGT) [START_REF] Yang | Life cycle reliability engineering[END_REF], Reliability Enhancement Test (RET) [START_REF] Benbow | The Certified Reliability Engineer Handbook[END_REF], Reliability Simulation Test (RST) [START_REF] Pecht | Physics-of-failure: an approach to reliable product development[END_REF][START_REF] Chen | Research on reliability simulation prediction of electronic product based on physisc of failure model[END_REF], etc. In this paper, we develop a new quantification method for the epistemic uncertainty in the adjustment factor method, based on the effectiveness of these engineering activities.

The contributions of this paper are summarized as follows: 1) a new reliability metric, the belief reliability, is developed to explicitly consider epistemic uncertainty in the model-based reliability methods; 2) a new method is developed to quantify epistemic uncertainty, based on the effectiveness of the engineering activities related to the reliability analysis and assessment of components and systems; 3) a method is developed to evaluate the belief reliability of components and systems, based on the integration of design margin, aleatory uncertainty and epistemic uncertainty. The rest of the paper is organized as follows. In section II, belief reliability is defined to account for the effect of epistemic uncertainty in model-based reliability methods. In section III-B, epistemic uncertainty is quantified based on the effectiveness of the related engineering activities and a belief reliability evaluation method is developed. Section IV presents two case studies to demonstrate the developed methods. Finally, the paper is concluded in section V with a discussion on future works.

II. DEFINITION OF BELIEF RELIABILITY

In this section, we introduce a new metric of reliability, belief reliability, to explicitly account for the influence of epistemic uncertainty on model-based reliability methods. We start with a brief introduction of the model-based reliability method in subsection II-A. Then, belief reliability is defined in subsection II-B.

A. Model-based reliability methods

For a general description of model-based reliability methods, we introduce the concepts of performance parameter and performance margin: Definition 1 (Performance parameter). Suppose failure occurs when a parameter p reaches a threshold value p th . Then, the parameter p is referred to as a performance parameter, while the threshold value p th is referred to as the functional failure threshold associated with p.

According to Definition 1, performance parameters and functional failure thresholds define the functional requirements on a system or a component, for which three categories exist in practice:

1) Smaller-the-better (STB) parameters: if failure occurs when p ≥ p th , then, the performance parameter p is a STB parameter. 2) Larger-the-better (LTB) parameters: if failure occurs when p ≤ p th , then, the performance parameter p is a LTB parameter. 3) Nominal-the-better (NTB) parameters: if failure occurs when p ≤ p th,L or p ≥ p th,U , then, the performance parameter p is a NTB parameter.

Definition 2 (Performance margin). Suppose p is a performance parameter and p th is its associated functional failure threshold; then,

m =                p th -p p th , if p is STB, p -p th p th , if p is LTB, min p th,U -p p th,U , p -p th,L p th,L , if p is NTB (1)
is defined as the (relative) performance margin associated with the performance parameter p.

Remark 1. From Definition 2, performance margin is a unitless quantity and failure occurs whenever m ≤ 0.

In the model-based reliability methods, it is assumed that the performance margin can be described by a deterministic model, which is derived based on knowledge of the functional principles and failure mechanisms of the component [START_REF] Collins | Mechanical design of machine elements and machines[END_REF][START_REF] Zeng | A physics-of-failurebased approach for failure behavior modeling: With a focus on failure collaborations[END_REF]. Conceptually, we assume that the performance margin model has the form

m = g m (x), (2)
where g m (•) denotes the deterministic model which predicts the performance margin and x is a vector of input variables.

In the design and manufacturing processes of a product, there are many uncertain factors influencing the input x of (2). Thus, the values of x may vary from product to product of the same type. Usually, this product-to-product variability is described by assuming that x is a vector of random variables with given probability density functions. Then, m is also a random variable and reliability R p is defined as the probability that m is greater than zero. The subscript p is used to indicate that R p is a probability measure. Given the probability density function of x, denoted by f X (•), R p can be calculated by:

R p = P r (g m (x) > 0) = • • • gm(x)>0 f X (x)dx. (3)

B. Definition of belief reliability

Belief reliability is defined in this subsection to explicitly account for the effect of epistemic uncertainty in model-based reliability methods. For this, we first define design margin and Aleatory Uncertainty Factor (AUF): Definition 3 (Design margin). Suppose the performance margin of a component or a system can be calculated by [START_REF] Meeker | Statistical methods for reliability data[END_REF]. Then, design margin m d is defined as

m d = g m (x N ), (4)
where x N is the nominal values of the parameters.

Definition 4 (Aleatory Uncertainty Factor (AUF)). Suppose R p is the probabilistic reliability calculated from the performance margin model using (3). Then, AUF σ m is defined as

σ m = m d Z Rp , (5)
where Z Rp is the value of the inverse cumulative distribution function of a standard normal distribution evaluated at R p .

Further, let equivalent design margin M E to be

M E = m d + m , (6)
where m ∼ Normal(0, σ 2 m ). It is easy to verify that M E ∼ Normal(m d , σ 2 m ) and R p can be calculated as the probability that M E > 0, as shown in Figure 1 (a). Therefore, the probabilistic reliability can be quantified by the equivalent performance margin and further by m d and σ m , where

• m d describes the inherent reliability of the product when all the input variables take their nominal values. Graphically, it measures the distance from the center of the equivalent performance margin distribution to the boundaries of the failure region, as shown in Figure 1 (a); • σ m accounts for the uncertainty resulting from the product-to-product random variations, e.g. the tolerance of manufacturing processes, the variability in material properties, etc. Usually, these random variations are controlled by engineering activities such as tolerance design, environmental stress screening, stochastic process control, etc [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF].

To further account for the effect of epistemic uncertainty, it is assumed that:

M E = m d + m + e , ( 7
)
where e is an adjustment factor [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF] and e ∼ Normal(0, σ 2 e ). Parameter σ e is defined as Epistemic Uncertainty Factor (EUF) and it quantifies the effect of epistemic uncertainty. The physical meaning of ( 7) is explained in Figure 1 (b): epistemic uncertainty introduces additional dispersion to the aleatory distribution of the equivalent performance margin. The degree of the dispersion is related to the knowledge we have on the failure process of the product, i.e., the more knowledge we have, the less value σ e takes. Considering the assumption made in [START_REF] Zio | The Monte Carlo simulation method for system reliability and risk analysis[END_REF], we can, then, define the belief reliability as follows:

Definition 5 (Belief reliability). The reliability metric

R B = Φ N m d σ 2 m + σ 2 e (8)
is defined as belief reliability, where Φ N (•) is the cumulative distribution function of a standard normal random variable.

Belief reliability can be interpreted as our belief degree on the product reliability, based on the knowledge of design margin, aleatory uncertainty and epistemic uncertainty. In the following, we discuss respectively how design margin, aleatory uncertainty and epistemic uncertainty influence the value of belief reliability. Discussion 1. It is obvious from (8) that R B ∈ [0, 1], where

• R B = 0 indicates that we believe for sure that a component or system is unreliable, i.e., it cannot perform its desired function under stated time period and operated conditions. • R B = 1 indicates that we believe for sure that a component or system is reliable, i.e., it can perform its desired function under stated time period and operated conditions. • R B = 0.5 indicates that we are most uncertain about the reliability of the component or system [START_REF] Liu | Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty[END_REF]. • R B,A > R B,B indicates that we believe that product A is more reliable than product B. Discussion 2 (Variation of R B with the design margin). From [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF], it is easy to see that R B is an increasing function of m d , as illustrated by Figure 2, which is in accordance with the intuitive fact that when the design margin is increased, the component or system becomes more reliable. Besides, it can be verified from ( 8) that if m d = 0, R B = 0.5. This is because when m d = 0, the product is at borderline between working and failure. Therefore, we are most uncertain about its reliability (For details, please refer to the maximum uncertainty principle in [START_REF] Liu | Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty[END_REF]).

Discussion 3 (Variation of R B with the aleatory uncertainty). In [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF], the effect of aleatory uncertainty is measured by the AUF, σ m . Figure 3 shows the variation of R B with σ m , when σ e is fixed, for different values of m d . It can be seen from Figure 3 that when m d and σ e are fixed, R B approaches 0.5 as σ m increases to infinity. The result is easy to understand, since σ m → ∞ indicates the fact that uncertainty has the greatest influence. In [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF], the effect of epistemic uncertainty is measured by the EUF, σ e . The variation of R B with respect to σ e is illustrated in Figure 4, with σ m fixed to 0.2. From Figure 4, we can see that when σ e → ∞, R B also approaches 0.5, for the same reason as the AUF. Besides, it can be shown from [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF] and assumption (3) that as σ e → 0, R B approaches the R p calculated by the modelbased reliability methods using equation ( 3). This is a natural result since σ e = 0 is the ideal case for which there is no epistemic uncertainty, so that the product failure behavior is accurately predicted by the deterministic performance margin model and the aleatory uncertainty.

In practice, we always have m d ≥ 0 and σ e > 0. Therefore,

R B ≤ R p (9)
where R p is the probabilistic reliability predicted by (3) under the same conditions. Equation [START_REF] Aven | A new perspective on how to understand, assess and manage risk and the unforeseen[END_REF] shows that using belief reliability yields a more conservative evaluation result than using the probabilistic reliability, because belief reliability considers the effect of insufficient knowledge on the reliability evaluations.

III. EVALUATION OF BELIEF RELIABILITY

In this section, we discuss how to evaluate the belief reliability for a given product. A general framework for belief reliability evaluation is first given in subsection III-A. Then, a method is presented for evaluating epistemic uncertainty and determining the value of the EUF.

A. Belief reliability evaluation

The R B defined in [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF] incorporates the contributions of design margin m d , aleatory uncertainty (represented by σ m ) and epistemic uncertainty (represented by σ e ). The contributions from the three factors should be evaluated individually and then, combined to evaluate the belief reliability of a component. Detailed procedures are presented in Figure 5.

Epistemic Uncertainty Evaluation

Performance Margin Model Development 2) Aleatory Uncertainty Evaluation: Next, the values of m d and σ m are determined. The value of m d is calculated based on [START_REF] Pecht | The reliability physics approach to failure prediction modelling[END_REF], where all the input parameters of the performance margin model take their nominal values. To calculated the value of σ m , the probabilistic reliability R p is calculated first by propagating aleatory uncertainty in the model parameters according to [START_REF] Aven | Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-probabilistic Methods[END_REF]. Either structural reliability methods [START_REF] Collins | Mechanical design of machine elements and machines[END_REF] or Monte Carlo simulations [START_REF] Zio | The Monte Carlo simulation method for system reliability and risk analysis[END_REF] might be used for the calculation. Then, σ m can be calculated by combining m d and R p using (5).

Aleatory Uncertainty Evaluation

3) Epistemic Uncertainty Evaluation: The value of σ e is, then, determined by evaluating the effect and potential impact of epistemic uncertainty. In practice, epistemic uncertainty relates to the knowledge on the component or system functions and failure behaviors: as this knowledge is accumulated, epistemic uncertainty is reduced. Hence, in this paper, we relate epistemic uncertainty to our state of knowledge on the product and its failure process and assess the value of σ e based on the effectiveness of engineering activities that generate our knowledge base. Details on how to evaluate the value of σ e is given in Section III-B.

4) Belief Reliability Evaluation: Following steps 1) -3), the values of m d , σ m and σ e are determined. Then, the belief reliability can be evaluated according to [START_REF] Zhang | Timedependent reliability analysis through response surface method[END_REF].

B. Quantification of epistemic uncertainty

In this section, we develop a method to quantify epistemic uncertainty based on the state of knowledge. In subsection III-B1, we discuss how to evaluate the state of knowledge, and then, in subsection III-B2, we quantify the effect of epistemic uncertainty in terms of σ e .

1) Evaluation of the state of knowledge: In the life cycle of a component or system, the knowledge on the products' failure behavior is gained by implementing a number of engineering activities of reliability analysis, whose purposes are to help designers better understand potential failure modes and mechanisms. In this paper, we refer to these engineering activities as epistemic uncertainty-related (EU-related) engineering activities. Table I lists some commonly encountered EU-related engineering activities and discusses their contributions to gaining knowledge and reducing epistemic uncertainty, where FMECA stands for Failure Mode, Effect and Criticality Analysis, FRACAS stands for Failure Reporting, Analysis, and Corrective Action System, RET stands for Reliability Enhancement Test, RGT stands for Reliability Growth Test and RST stands for Reliability Simulation Test.

In this paper, we make an assumption that the state of knowledge is directly related to the effectiveness of the EU-related engineering activities. Suppose there are n EUrelated engineering activities in a product life cycle. Let y i , i = 1, 2, • • • , n denote the effectiveness of the EU-related engineering activities, where y i ∈ [0, 1]; the more effective the engineering activity is, the larger value the corresponding y i takes. The values of y i are determined by asking experts to evaluate the effectiveness of the EU-related engineering activities, based on a set of predefined evaluation criteria.

For example, the effectiveness of FMECA can be evaluated based on eight elements, as shown in Table II. For each element, experts are invited to evaluate their performances according to the criteria listed in Table II. Based on the evaluated performance, a score can be assigned to each element, denoted by S 1 , S 2 , • • • , S 8 . Then, the effectiveness of FMECA, denoted by y 1 , can be determined by

y 1 = 1 8 8 i=1 S i . ( 10
)
The effectiveness of other EU-related engineering activities can be evaluated in a similar way, so that the values

Activities

Contributions to gaining knowledge and reducing epistemic uncertainty FMECA FMECA helps designers to identify potential failure modes and understand their effects, so as to increase the designer's knowledge about potential failures [START_REF] Carlson | Effective FMEAs: Achieving safe, reliable, and economical products and processes using failure mode and effects analysis[END_REF].

FRACAS

By implementing FRACAS, knowledge on potential failure modes and mechanisms is accumulated based on previously ocurred failures and corrective actions [START_REF] Benbow | The Certified Reliability Engineer Handbook[END_REF].

RGT

In a RGT, cycles of Test Analysis and Fix (TAAF) are repeated until the product reaches its reliability requirements. In this way, designers' knowledge on the failure modes and mechanisms is accumulated [START_REF] Yang | Life cycle reliability engineering[END_REF].

RET

As the RGT, RET reduces epistemic uncertainty by stimulating potential failures, but using highly accelerated stresses, which can generate failures that are hard to be identified by analyses or conventional tests [START_REF] Benbow | The Certified Reliability Engineer Handbook[END_REF].

RST

In a RST, simulation tests are conducted based on physics-of-failure models to identify weak design points for the products. Knowledge of potential failure modes can be accumulated in this way [START_REF] Pecht | Physics-of-failure: an approach to reliable product development[END_REF][START_REF] Chen | Research on reliability simulation prediction of electronic product based on physisc of failure model[END_REF]. Definition of failures The failure definition is clear and unambiguous, so that a complete analysis of failure modes could be conducted.

S 1 = 3
The failure definition is defined, but unclear or ambiguous. S 1 = 1 The failures are undefined.

1 = 0 S 2
Coverage of failure modes The considered failure modes cover all that have occurred historically in the products with similar functions, use and environmental conditions.

S 2 = 3
A few uncritical failure modes are not considered in the analysis. S 2 = 1 A lot of critical failure modes are not considered in the analysis.

S 2 = 0 S 3
Completeness of failure mode analysis The considered failure modes include both catastrophic failures as well as degradation failures.

S 3 = 3
Only one of the two types is considered. S 3 = 1 None of the two types is considered.

S 3 = 0 S 4
Credibility of information sources The considered failure modes come from real historical data of the product or similar products.

S 4 = 3
The considered failure modes come from literature. S 4 = 1 The considered failure modes come from expert judgements.

S 4 = 0 S 5
Completeness of failure cause analysis The analysis takes into account all the possible failure causes. S 5 = 3 A few noncritical failure causes are not considered in the analysis.

S 5 = 1 A lot of critical failure causes are not considered in the analysis.

S 5 = 0 S 6
Completeness of failure effect analysis Failure effects in both component and system levels are analyzed. S 6 = 3 Only one of the two levels is considered.

S 6 = 1 Failure effects are not considered in the analysis.

S 6 = 0 S 7
Credibility of data sources Criticality analysis is based on field data. S 7 = 3 Criticality analysis is based on data from literature.

S 7 = 1 Criticality analysis is based on data from expert judgements.

S 7 = 0 S 8
Effectiveness of design improvements Over 90% of the exposed design weaknesses are modified. S 8 = 3 The modified design weaknesses are between 60%-90%.

S 8 = 1 Less than 60% of the exposed design weaknesses are modified.

S 8 = 0 for y 1 , y 2 , • • • , y n can be determined. Then, the state of knowledge about the potential failures of the component or system can be evaluated as the weighted average of

y i , i = 1, 2, • • • , n: y = n i=1 ω i y i , (11)
where ω i is the relative importance of the ith engineering activity for the characterization of the potential failure behaviors, where

n i=1 ω i = 1.
2) Determination of EUF: Having determined the value of y, we need to define a function σ e = h(y), through which σ e is determined. Since σ e is a measure of the severity of epistemic uncertainty and y measures the state of knowledge, σ e is negatively dependent on y. Theoretically, any monotonic decreasing function of y could serve as h(y). In practice, the form of h(y) reflects the decision maker attitude towards epistemic uncertainty and is related to the complexity of the product. Therefore, we propose h(y) to be

h(y) =                1 3 √ y • m d , for simple products; 1 3y 6 • m d , for complex products; 1 3y 2 • m d , for medium complex products. ( 12
) By letting σ m = 0 and m d fixed to a constant value, the attitudes of the decision maker for different products can be investigated (see Figure 6):

• for simple products, R B is a convex function of y, indicating that even when y is small, we can gather enough knowledge on the product function and failure behaviors, so that we can assign a high value to the belief reliability; • for complex products, R B is a concave function of y, indicating that only when y is large we can gather sufficient knowledge on the product function and failure behaviors, so that we can assign a high value to the belief Fig. 6. Different attitudes of the decision maker towards epistemic uncertainty reliability; • the h(y) for medium complex products lies between the two extremes.

IV. CASE STUDIES

In this section, we apply the developed belief reliability to evaluate the reliability of two engineering components/systems, i.e., a Hydraulic Servo Actuator (HSA) in Section IV-A and a Single Board Computer (SBC) in Section IV-B. A comparison is also made on both cases with respect to the traditional probabilistic reliability metrics.

A. Hydraulic Servo Actuator (HSA)

The HSA considered in this paper comprises the six components, as listed in Table III. The schematic of the HSA is given in Figure 7.

p HSA = -20 lg A HC A obj , (13)
where, A HC denotes the amplitude of the HC displacements when input signal x input is a sinusoidal signal, and A obj is the objective value of A HC . Failure occurs when p HSA ≥ p th = 3(dB). The belief reliability of the HSA is evaluated following the procedures in Figure 5.

1) Performance Margin Model Development:

The performance margin model is developed in two steps. First, a model for the p HSA is developed based on hydraulic principles, with the help of commercial software AMESim. The AMESim model is given in Figure 7. Coherently with (2), the model in Figure 7 is written as

p HSA = g HSA (x HSA ). ( 14
)
Second, as p HSA is a STB performance parameter, the performance margin of the HSA can be determined according to [START_REF] Ebeling | An introduction to reliability and maintainability engineering[END_REF]:

m HSA = 1 p th (p th -g HSA (x HSA )) . (15)
2) Aleatory Uncertainty Evaluation: The x HSA comprises six parameters, namely, the clearances on diameters (CoDs) of the six components of the HSA. The CoDs are subject to aleatory uncertainties from production and manufacturing processes, which are quantified by the tolerances in Table III. For simplicity of illustration, it is assumed that all the six parameters follow normal distributions. Following the '3σ' principle (for references, see [40]), the probability density function for each parameter is determined and given in Table III. The value of m d is calculated by [START_REF] Pecht | The reliability physics approach to failure prediction modelling[END_REF], where the nominal values are given in Table III. The resulting m d is 0.6928 (dB). The values of σ m is determined using Monte Carlo simulations with a sample size N = 3000. The resulting σ m is 0.0353 (dB).

3) Epistemic Uncertainty Evaluation: Then, we need to determine the value of σ e . In the development of the HSA, five EU-related engineering activities, i.e., FMECA, FRA-CAS, RGT, RET and RST have been conducted. Let y i , i = 1, 2, • • • , 5 denote the five engineering activities, respectively. The values of y i s can be determined by evaluating the effectiveness of these engineering activities, based on the procedures illustrated in Section III-B1. The result is y 1 = 0.70, y 2 = 0.90, y 3 = 0.80, y 4 = 0.85, y 5 = 0.70. In this case study, the engineering activities are assumed to have equal weights, ω 1 = ω 2 = • • • = ω 5 = 1/5, and then, according to [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF], y = 0.79. Since the HSA has medium complexity, according to [START_REF] Aven | Model output uncertainty in risk assessment[END_REF],

σ e = 1 3y 2 • m d = 0.3700. ( 16
)
4) Belief Reliability Evaluation: Finally, the belief reliability can be predicted using ( 8) and the result is shown in Table IV. If we only consider the aleatory uncertainty, probabilistic reliability can be predicted using (3), whose value is also presented in Table IV for comparisons. The result shows that, ESV: Electrohydraulic servo valve HC: Hydraulic cylinder as expected, epistemic uncertainty reduces our confidence that the product will perform its function as designed, whereas probabilistic reliability would lead to overconfidence. Another major difference between belief reliability and probabilistic reliability is that belief reliability allows for the consideration of EU-related engineering activities in the reliability assessment, which are neglected in the probabilitybased reliability evaluation. For example, if the effectiveness of the EU-related engineering activities is increased from y 1 = 0.70, y 2 = 0.90, y 3 = 0.80, y 4 = 0.85, y 5 = 0.70 to y 1 = y 2 = • • • = y 5 = 0.9, then, the belief reliability will increase from R B,0 = 0.9688 to R B,1 = 0.9921. In other words, in order to enhance the belief reliability, one not only needs to increase the design margin and reduce aleatory uncertainty by design, but also needs to reduce epistemic uncertainty by improving the state of knowledge, whereas probabilistic reliability focuses only on the former two aspects.

B. Single Board Computer

A SBC, as shown in Figure 8 [41], is chosen to demonstrate the time-dependent belief reliability analysis for electrical systems. A probabilistic reliability analysis was conducted in [41] based on the parts-counting reliability prediction method in [42]. The times to failure of both the components are assumed to be exponentially distributed and their failure rates are predicted based on the database in [42], as shown in Table V. The failure rate of the SBC can, then, be calculated by summing over all the components' failure rates. Hence, the predicted probabilistic reliability is

R p (t) = exp{-1.186 × 10 -6 t}, (17)
where the unit of t is hour. The probabilistic reliability in ( 17) is a time-dependent function. To further evaluate the belief reliability, first note by substituting (5) into (8), we have

R B = 1 1 Z Rp 2 + σ e m d 2 . ( 18
)
Since R p is time-dependent, the belief reliability is also a time-dependent function and can be calculated by using [START_REF] Droguett | Bayesian methodology for model uncertainty using model performance data[END_REF] recursively at each time t:

R B (t) = 1 1 Z Rp(t) 2 + σ e m d 2 , ( 19
)
where R p (t) is the time-dependent probabilistic reliability and σ e is the EUF evaluated using the procedures in Section III-B. The effectiveness of the five EU-related engineering activities, i.e., FMECA, FRACAS, RGT, RET and RST, can be assessed using the procedures illustrated in Section III-B1: y 1 = 0.60, y 2 = 0.80, y 3 = 0.70, y 4 = 0.75, y 5 = 0.55. As the previous case study, we also assume that the five activities have equal weights. From [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF], y = 0.68. By assessing the configuration of the SBC, it is determined that it has medium complexity. Therefore, by substituting ( 12) and ( 17) into [START_REF] Apostolakis | The concept of probability in safety assessments of technological systems[END_REF], the belief reliability of the SBC can be calculated, as shown in Figure 9. It can be seen from Figure 9 that the belief reliability curve is more close to R B = 0.5 than the probabilistic reliability. This is because R B = 0.5 corresponds to the state of maximum uncertainty, since we cannot differentiate whether the system is more likely to be working or failure (for details, please refer to maximum uncertainty principle in [START_REF] Liu | Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty[END_REF]). Since belief reliability considers the influence of epistemic uncertainty, it yields a more uncertain result than the probabilistic reliability.

A sensitivity analysis is conducted with respect to y to further investigate the influence of epistemic uncertainty on belief reliability. The results are given in Figure 10. It can be seen from Figure 10 that the value of y significantly impacts R B : a larger value of y, which indicates improvements on the effectiveness of the EU-related engineering activities, tends to make the belief reliability moving towards the probabilistic reliability; while a lower value of y tends to make the belief reliability moving towards 0.5, which is the state of maximum uncertainty. This demonstrates that, compared to the traditional probabilistic reliability, belief reliability allows for the explicit consideration of epistemic uncertainty and EUrelated engineering activities in the reliability assessment. In other words, in order to enhance the belief reliability, one not only needs to increase the design margin and reduce aleatory uncertainty by design, but also needs to reduce epistemic uncertainty by improving the state of knowledge.

V. CONCLUSION

In this paper, a new metric of belief reliability has been introduced to explicitly incorporate the influence of epistemic uncertainty into model-based methods of reliability assessments. To quantify the effect of epistemic uncertainty, an evaluation method is proposed, based on the effectiveness of engineering activities related to reliability analysis and assessment. The proposed belief reliability evaluation method integrates design margin, aleatory uncertainty and epistemic uncertainty for a comprehensive and systematic characterization of reliability. Two numerical case studies demonstrate the benefits of belief reliability compared to the traditional probability-based reliability metrics, with the explicit consideration of epistemic uncertainty.

Compared to the traditional probabilistic reliability metrics, belief reliability explicitly considers the effect of epistemic uncertainty and allows considering EU-related engineering activities in reliability assessment. We believe that as a new reliability metric, belief reliability is beneficial in reliability engineering practices, since epistemic uncertainty is a severe problem for real-world products, especially for those in design and development phases. An interesting future work is to define a mathematical theory to model belief reliability and its time-dependence. Various mathematical theories dealing with epistemic uncertainty can be considered, e.g., Bayesian theory, evidence theory, possibility theory, uncertainty theory, etc. Besides, methods of scoring the effectiveness of engineering activities should be further investigated.

Fig. 1 .

 1 Fig. 1. Epistemic uncertainty effect on the distribution of the equivalent performance margin

Fig. 2 .

 2 Fig. 2. Influence of m d on R B

Fig. 3 .

 3 Fig. 3. Variation of R B with σm

Fig. 4 .

 4 Fig. 4. Variation of R B with σe

Fig. 5 .

 5 Fig. 5. Procedures for component belief reliability evaluation Four steps comprise the evaluation procedure: 1) Performance Margin Model Development: First, a deterministic performance margin model is developed to predict the value of the performance margin m. The performance margin model can be developed based on knowledge of underlying functional principles and physics of failures. For a detailed discussion on how to develop performance margin models, readers might refer to [38] and [39].2) Aleatory Uncertainty Evaluation: Next, the values of m d and σ m are determined. The value of m d is calculated based on[START_REF] Pecht | The reliability physics approach to failure prediction modelling[END_REF], where all the input parameters of the performance margin model take their nominal values. To calculated the value of σ m , the probabilistic reliability R p is calculated first by propagating aleatory uncertainty in the model parameters according to[START_REF] Aven | Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-probabilistic Methods[END_REF]. Either structural reliability methods[START_REF] Collins | Mechanical design of machine elements and machines[END_REF] or Monte Carlo simulations[START_REF] Zio | The Monte Carlo simulation method for system reliability and risk analysis[END_REF] might be used for the calculation. Then, σ m can be calculated by combining m d and R p using (5).

Fig. 7 .

 7 Fig. 7. Schematic of the AMESim model to predict p HSA The required function of the HSA is to transform input electrical signals, x input , into the displacement of the hydraulic

Fig. 8 .

 8 Fig. 8. A SBC [41]

Fig. 9 .

 9 Fig. 9. Belief reliability of the SBC

TABLE I EXAMPLES

 I OF EU-RELATED ENGINEERING ACTIVITIES

  DistributionsN(1, 0.005 2 ) N(7, 0.05 2 ) N(7, 0.05 2 ) N(7, 0.05 2 ) N(7, 0.05 2 ) N(10, 0.5 2 )

				TABLE III
			COMPONENTS AND TOLERANCES OF THE HSA
	Component	ESV	Spool 1	Spool 2	Spool 3	Spool 4	HC
	Parameters	CoD x 1	CoD x 2	CoD x 3	CoD x 4	CoD x 5	CoD x 6
	Tolerances (×10 -3 mm)	1 ± 0.015	7 ± 0.15	7 ± 0.15	7 ± 0.15	7 ± 0.15	10 ± 1.5

ACKNOWLEDGMENT This work has been performed within the initiative of the Center for Resilience and Safety of Critical Infrastructures (CRESCI, http://cresci.cn). The research of Dr. Zhiguo Zeng is partially supported by Natural Science Foundation of China (NSFC) under grant No. 71601010. The research of Dr. Meilin Wen is supported by Natural Science Foundation of China (NSFC) under grant No. 71671009. The research of Prof. Rui Kang is supported by Natural Science Foundation of China (NSFC) under grant No. 61573043.

2010.

[38] Z. Zeng, R. Kang, and Y. Chen, "Using pof models to predict system reliability considering failure collaboration," Chinese Journal of Aeronautics, vol. 29, no. 5, pp. 1294-1301, 2016.

[39] Z. Zeng, Y. Chen, E. Zio, and R. Kang, "A compositional method to model dependent failure behavior based on pof models," Chinese Journal of Aeronautics, 2017. His research focuses on the characterization and modeling of the failure/repair/maintenance behavior of components, complex systems and critical infrastructures for the study of their reliability, availability, maintainability, prognostics, safety, vulnerability and security, mostly using a computational approach based on advanced Monte Carlo simulation methods, soft computing techniques and optimization heuristics. He is author or co-author of five international books and more than 170 papers on international journals.


PhD Zhiguo Zeng
email: zhiguo.zeng@centralesupelec.fr

Professor Rui Kang
email: kangrui@buaa.edu.cn

Assistant professor b Meilin Wen
email: wenmeilin@buaa.edu.cn

Professor Enrico Zio
email: enrico.zio@ecp.fr

Uncertainty theory as a

Keywords: Reliability, epistemic uncertainty, uncertainty theory, belief reliability, fault tree

published or not. The documents may come

Introduction

Modern reliability engineering is increasingly looking at the model-based methods (cf. physics-of-failure (PoF) methods [START_REF] Chamberlain | Development of a probabilistic mechanistic model for reliability assessment of gas cylinders in compressed natural gas vehicles[END_REF], structural reliability methods [START_REF] Collins | Mechanical design of machine elements and machines[END_REF], etc.), where reliability is predicted exploiting deterministic failure behavior models whose parameter variations are assumed to be the only source of uncertainty [START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. In practice, however, apart from the random variations in the model parameters (often referred to as aleatory uncertainty [1]), the predicted reliability is also subject to the influence of epistemic uncertainty due to incomplete knowledge on the degradation and failure processes [START_REF] Kiureghian | Aleatory or epistemic? does it matter?[END_REF]: for example, the developed failure behavior model might not be able to accurately describe the actual failure process; besides, the precise values of the model parameters might not be accurately estimated [START_REF] Aven | Model output uncertainty in risk assessment[END_REF][START_REF] Bjerga | An illustration of the use of an approach for treating model uncertainties in risk assessment[END_REF], etc. In most existing model-based reliability assessment methods, however, the effect of epistemic uncertainty has not been considered.

Recently, a new metric of reliability, the belief reliability, has been defined to explicitly account for epistemic uncertainty in model-based reliability analysis and assessment [START_REF] Fan | An approach to measure reliability based on belief reliability[END_REF][START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF][START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. The new reliability metric integrates the contributions of design margin, aleotory uncertainty and epistemic uncertainty and provides a more comprehensive and systematic description of reliability. Zeng et al. [START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF] presented a framework to evaluate the belief reliability where epistemic uncertainty is quantified by the effectiveness of the engineering analysis and assessment activities that contribute to the state of knowledge on the failure causes and processes. Belief reliability has been applied successfully on the reliability evaluation of hydraulic servo-actuators [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF][START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF], DC regulated power supplies [START_REF] Fan | An approach to measure reliability based on belief reliability[END_REF] and printed circuit boards [START_REF] Jiang | A naive bayes based method for evaluating the effects of electronic product reliability simulation experiments[END_REF], all of which are subject to the influence of epistemic uncertainty.

Currently, the belief reliability of a component or a system can only be evaluated from its definition (i.e., based on design margin, aleotory uncertainty and epistemic uncertainty) [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF]. In practice, we often need to calculate the belief reliability of a system based on the structure of the system and the belief reliabilities of its components (referred to as system reliability analysis in conventional reliability theories [START_REF] Yang | Life cycle reliability engineering[END_REF]). To address this problem, a mathematical theory should be determined as the mathematical foundation of belief reliability, based on which the system belief reliability analysis method can be developed.

In literature, various mathematical theories have been used to describe epistemic uncertainty, e.g., probability theory (subjective interpretation [START_REF] De Finetti | Probability, induction, and statistics[END_REF]), evidence theory [START_REF] Shafer | A mathematical theory of evidence[END_REF], possibility theory [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF] and uncertainty theory [START_REF] Liu | Uncertainty theory[END_REF], etc. Kang et al. [START_REF] Kang | Measuring reliability under epistemic uncertainty: A review on non-probabilistic reliability metrics[END_REF] reviewed the theories and concluded that among them, uncertainty theory is the most suitable one for modeling belief reliability since it satisfies the Duality Axiom and adopts minimum operation as the Product Axiom, which are two essential requirements for a mathematical theory qualified to describe reliability under the influence of epistemic uncertainty. If either requirement is violated, misleading results might be reached when belief reliability is applied in practical applications (see Section 3.2 for a detailed discussion).

Uncertainty theory, proposed by Liu in 2007 [START_REF] Liu | Uncertainty theory[END_REF] and refined by Liu in 2010 [START_REF] Liu | Uncertainty theory: A branch of mathematics for modeling human uncertainty[END_REF], is a branch of axiomatic mathematic founded on four axioms, the Normality, Duality, Subadditivity and Product Axiom. Currently, uncertainty theory has been widely applied in various fields, including portfolio selection [START_REF] Zhang | Uncertain programming models for portfolio selection with uncertain returns[END_REF], network science [START_REF] Han | The maximum flow problem of uncertain network[END_REF], option pricing [START_REF] Ji | Option pricing for an uncertain stock model with jumps[END_REF], graph theory [START_REF] Gao | On distribution function of the diameter in uncertain graph[END_REF], transportation [START_REF] Yang | Reduction methods of type-2 uncertain variables and their applications to solid transportation problem[END_REF],

supply chain [START_REF] Huang | Model and algorithm for 4plrp with uncertain delivery time[END_REF], etc. The research of reliability in uncertainty theory started from [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF], where Liu defined the reliability index and showed how to calculate the system reliability index from the system structure functions. In [START_REF] Liu | Reliability analysis of redundant system with uncertain lifetimes[END_REF], the reliability indexes for redundant systems were calculated for the case in which the lifetimes of the components are uncertain variables. Zeng et al. [START_REF] Zeng | Belief reliability: a new metrics for products' reliability[END_REF] defined time-static and time-variant reliability in the context of uncertainty theory and developed calculation methods for the reliability indexes. Wen and Kang [START_REF] Wen | Reliability analysis in uncertain random system[END_REF] developed an approach to calculate the reliability index when both uncertain variables and random variables are considered. Gao and Yao [START_REF] Gao | Importance index of components in uncertain random systems[END_REF] investigated the importance index in the context of uncertainty theory. Age replacement and block replacement policies were also investigated with lifetimes described as uncertain variables [START_REF] Ke | Block replacement policy with uncertain lifetimes[END_REF][START_REF] Zhang | Some new results on uncertain age replacement policy[END_REF][START_REF] Zhang | Uncertain block replacement policy with no replacement at failure[END_REF].

Most existing system reliability analysis methods in uncertainty theory are based on structure functions (e.g., see [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF] and [START_REF] Wen | Reliability analysis in uncertain random system[END_REF]). Since they require enumerating all the possible combination of system states, the computational efficiency of the structure function-based methods are often unsatisfactory, especially for large and complex systems. In a previous study, minimal cut sets have been used to alleviate the computational burdens of the structure function-based methods [START_REF] Zeng | Belief reliability: a new metrics for products' reliability[END_REF]. However, the method developed in [START_REF] Zeng | Belief reliability: a new metrics for products' reliability[END_REF] requires independence among the minimal cut sets, which is a strong condition and restricts its application.

In this paper, we show that the restriction is unnecessary and develop a minimal cut set-based method to calculate the belief reliability for a system with independent components. The rest of this paper is organized as follows. Section 2 reviews the definition of belief reliability. In Section 3, we justify the choice of uncertainty theory as the mathematical foundation of belief reliability and give the definition of belief reliability in the context of uncertainty theory. Then, a system belief reliability analysis method is developed based on minimal cut sets in Section 4. In Section 5, a numerical algorithm is presented for belief reliability analysis based on fault tree models. The paper is concluded in Section 6 with discussions on possible future research directions.

Definition of belief reliability

In traditional model-based reliability methods, it is assumed that the failure behavior of a component or system is characterized by its performance margin m, which is modeled by:

m = g m (x), (1)
where m ≤ 0 indicates that the component or system fails and m > 0 indicates normal functioning; g m (•) is developed by modeling the failure process [START_REF] Zeng | Using pof models to predict system reliability considering failure collaboration[END_REF].

Given the probability density functions of the input variables x, denoted by f X (x), the reliability index can be calculated as

R p = P r (g m (x) > 0) = • • • gm(x)>0 f X (x)dx. ( 2
)
To differentiate it from belief reliability, the reliability index in (2) is referred to as probabilistic reliability in this paper.

In the model-based reliability methods, a fundamental assumption is that, the reliability model is correct and accurate, so that all the uncertainty comes from the random variations in x (aleatory uncertainty). The validity of such an assumption heavily depends on the state-of-knowledge we have on the failure process. In a lot of practical applications, however, due to the limitation of the knowledge, the models in ( 1) and ( 2) might not be able to accurately capture the actual failure process. Besides, the precise values of the model parameters might not be accurately known to us. Therefore, the predicted reliability index is subject to an additional source of uncertainty, which arises from lack of knowledge and is referred to as epistemic uncertainty [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF].

Belief reliability was proposed as a metric of reliability that explicitly accounts for epistemic uncertainty in reliability analysis and assessment [START_REF] Fan | An approach to measure reliability based on belief reliability[END_REF][START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF][START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. Note that in (1) and ( 2), the probabilistic reliability R p can be viewed as determined by deterministic designs and aleatory uncertainty in the design parameters. Deterministic designs are quantified by design margin m d :

m d = g m (x N ) (3)
where x N is the nominal values of the parameters. Aleatory uncertainty is measured by F a , the factor of aleatory uncertainty, which is defined by:

F a = m d Z Rp (4)
where R p is given by ( 1) and (2); Z α is the value of the inverse cumulative distribution function of a standard normal distribution evaluated at α. Let us define equivalent performance margin M E as:

M E = m d + a , (5)
where m d is the design margin in (3) and a ∼ Normal(0, F 2 a ) quantifies the effect of aleatory uncertainty. It is easy to verify that

M E ∼ Normal(m d , F 2 a )
and the probabilistic reliability R p can be calculated as the probability that M E > 0, as shown in Figure 1 (a).

In belief reliability, epistemic uncertainty is described by introducing a factor of epistemic uncertainty, denoted by F e , whose value is related to the state-ofknowledge of the failure processes and is measured based on the effectiveness of the engineering analysis and assessment activities for component and system reliability performance characterization [START_REF] Fan | An approach to measure reliability based on belief reliability[END_REF][START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. An adjustment factor e ∼ Normal(0, F 2 e ) is introduced to quantify the effect of epistemic uncertainty on the equivalent performance margin:

M E = m d + a + e . (6)
Equation ( 6) indicates that epistemic uncertainty introduces additional dispersion to the aleatory distribution of the equivalent performance margin, as shown in Figure 1 (b). Considering ( 6) and the normality assumption on a and e , belief reliability is defined as:

Definition 1 (Belief reliability [START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]). The reliability metric

R B = Φ N m d F 2 a + F 2 e ( 7
)
is defined as belief reliability, where Φ N (•) is the cumulative distribution function of a standard normal random variable.

It can be shown from ( 7) that as F e → 0, R B → R p , where R p denotes the conventional model-based reliability metric calculated under the same conditions. This is natural, since F e → 0 indicates that there is no epistemic uncertainty and, therefore, the failure behavior can be accurately determined by the reliability models in ( 1) and ( 2).

In practical application, we always have m d > 0 and F e ≥ 0 [START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. It is easy to verify from ( 7) that

R B ≤ R p , (8)
which shows that using belief reliability yields a more conservative evaluation result than using the probability-based reliability metric. The reason is that belief reliability considers the effect of insufficient knowledge on the estimated reliability, while the probability-based reliability metric implicitly assumes that knowledge is complete. It is the additional uncertainty caused by the insufficient knowledge that reduces our confidence on the reliability estimation.

Uncertainty theory as the mathematical foundation of belief reliability

In this section, we discuss the mathematical foundations of belief reliability and show that the new reliability metric should be modeled by uncertainty theory. Uncertainty theory is reviewed in subsection 3.1. In subsection 3.2, we explain the reasons to choose uncertainty theory as the mathematical foundation, and then define belief reliability as an uncertain measure.

Preliminaries of Uncertainty Theory

The first important concept in uncertainty theory is that of an event. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ in L is called an event.

In uncertainty theory, the belief degree of an event is measured by its uncertain measure. An uncertain measure is a set function M from L to [0, 1] satisfying the following three axioms [START_REF] Liu | Uncertainty theory[END_REF]:

Axiom 1 (Normality Axiom [START_REF] Liu | Uncertainty theory[END_REF]). M{Γ} = 1 for the univeral set Γ.

Axiom 2 (Duality Axiom [START_REF] Liu | Uncertainty theory[END_REF]). M{Λ} + M{Λ c } = 1 for any event Λ.

Axiom 3 (Subadditivity Axiom [START_REF] Liu | Uncertainty theory[END_REF]). For every countable sequence of events

Λ 1 , Λ 2 , • • • , M ∞ i=1 Λ i ≤ ∞ i=1 M{Λ i }. (9)
The triplet (Γ, L, M) is called an uncertainty space [START_REF] Liu | Uncertainty theory[END_REF]. A product uncertain measure was defined by Liu [START_REF] Liu | Some research problems in uncertainty theory[END_REF] in order to obtain an uncertain measure of a compound event, thus producing the fourth axiom of uncertainty theory:

Axiom 4 (Product Axiom [START_REF] Liu | Some research problems in uncertainty theory[END_REF]).

Let (Γ k , L k , M k ) be uncertainty spaces for k = 1, 2, • • • The product uncertain measure M is an uncertain measure satisfying M ∞ k=1 Λ k = ∞ k=1 M k {Λ k } ( 10
)
where Λ k are arbitrarily chosen events from

L k for k = 1, 2, • • • , respectively.
An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e. , for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B} is an event [START_REF] Liu | Uncertainty theory[END_REF].

In practice, an uncertain variable is described by the uncertainty distribution [START_REF] Liu | Uncertainty theory[END_REF], defined by

Φ(x) = M{ξ ≤ x}, ∀x ∈ . ( 11
)
An uncertainty distribution is said to be regular if its inverse function Φ -1 (•) exists and is unique for each α ∈ (0, 1) [START_REF] Liu | Uncertainty theory: A branch of mathematics for modeling human uncertainty[END_REF].

The uncertain variables ξ 1 , ξ 2 , . . . , ξ m are said to be independent if

M m i=1 (ξ i ∈ B i ) = m i=1 M{ξ i ∈ B i } (12)
for any Borel sets B 1 , B 2 , • • • , B m of real numbers [START_REF] Liu | Some research problems in uncertainty theory[END_REF].

Liu [START_REF] Liu | Uncertainty theory: A branch of mathematics for modeling human uncertainty[END_REF] developed operation laws for uncertain variables so that the distribution of functions of independent uncertain variables can be achieved. Let

ξ 1 , ξ 2 , • • • , ξ n be independent uncertain variables with regular uncertainty distri- butions Φ 1 , Φ 2 , • • • , Φ n , respectively. If the function f (x 1 , x 2 , • • • , x n ) is strictly increasing with respect to x 1 , x 2 , • • • , x m ,
and strictly decreasing with respect

to x m+1 , x m+2 , • • • , x n , then, the uncertain variable ξ = f (ξ 1 , ξ 2 , • • • , ξ n ) has an inverse uncertainty distribution Ψ -1 (α) = f Φ -1 1 (α), Φ -1 2 (α), • • • , Φ -1 m (α), Φ -1 m+1 (α), • • • , Φ -1 n (α) . (13)

Belief reliability as an uncertain measure

Belief reliability measures the degree to which we believe that a component or a system can perform its function as designed. In this subsection, we compare four mathematical theories commonly used to model belief degrees, probability theory (subjective interpretation [START_REF] De Finetti | Probability, induction, and statistics[END_REF]), evidence theory [START_REF] Shafer | A mathematical theory of evidence[END_REF], possibility theory [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF] and uncertainty theory [START_REF] Liu | Uncertainty theory[END_REF], and choose among them the most appropriate one as the mathematical foundation for belief reliability.

In practice, how to calculate the belief degree of the intersection of events (more formally, the product event) is an important issue, since it is the basis of system reliability calculations. Based on how the belief degree of the intersection of events is calculated, the four theories can be divided into two groups.

Probability theory and evidence theory comprise the first group, where the belief degree of the intersection of events is calculated by the product of the individual belief degrees (assuming independence among the individual events).

According to Liu [START_REF] Liu | Why is there a need for uncertainty theory[END_REF], a premise of using the product operation to calculate the belief degree of the intersection of events is that the estimated belief degree for each individual event is close enough to the long-run cumulative frequency.

As shown in [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF], however, belief reliability is a more conservative reliability measure than the probabilistic reliability. If we use probability theory or evidence theory to model belief reliability, the conservatism in the component level will be distorted by the product operation, which might lead to counter-intuitive results when calculating system belief reliability. To illustrate this point, consider the following example.

Based on the evaluation result in (14), the system is highly unreliable, which contradicts with our intuition.

Example 1 shows that to model belief reliability, we need a mathematical theory whose operation law of product events can compensate for the conservatism in the component-level belief reliability evaluation. Possibility theory provides an alternative solution by assuming that the product belief degree is the minimum one among all the individual events [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF][START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF]. If we regard the component belief reliabilities in Example 1 as a possibility measure, according to [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF], the system belief reliability is given by

R B,S = 2000 i=1 R B,i = 0.9987, (15)
which avoids the counter-intuitive result in Example 1. However, regarding belief reliability as a possibility measure introduces an issue: possibility measure does not follow the duality axiom, which might lead to other counter-intuitive results [START_REF] Liu | Uncertainty theory: A branch of mathematics for modeling human uncertainty[END_REF]. For instance, see Example 2.

Example 2. Assume that belief reliability R B is a possibility measure. A possibility measure Π has the following properties [START_REF] Dubois | Possibility theory: An approach to the computerized processing of information[END_REF]:

• Π(Ω) = 1,
where Ω is the universal set, and

• Π(U ∪ V ) = Π(U ) ∨ Π(V )
, for any pair of disjoint sets U and V .

Since "working" and "failure" are two disjoint sets and their union is the universal set, from the above axioms, it is easy to show that for a given component or a system, either the reliability R B = 1 or the unreliability R B = 1 which will confuse the decision maker when applied in practice.

From

R B = M{Λ 1 }. ( 16
)
Remark 1. From the Duality Axiom, we can calculate the belief unreliability:

R B = M{Λ 2 } = 1 -R B , (17)
which can also be seen from Figure 1, since the areas of failure region and safe region sum up to 1.

Minimal Cut Set Theorem

In this section, we show how to calculate the belief reliability of a coherent system by proving the Minimal Cut Set Theorem. Coherent system is the most widely applied system model in reliability theory, which describes the logic of binary monotone systems whose components are all relevant [START_REF] Bayramoglu | The reliability of coherent systems subjected to marshall-olkin type shocks[END_REF][START_REF] Navarro | Orderings of coherent systems with randomized dependent components[END_REF].

Commonly encountered examples of coherent systems include series systems, parallel systems, k-out-n:G systems, etc.

Let ξ i , 1 ≤ i ≤ n and ξ denote the state of the ith component and of the system, respectively, where

ξ i =      1, if the ith component is working, 0, if the ith component fails. ξ =      1, if the system is working, 0, if the system fails. ( 18
)
The boolean variables ξ and ξ i , 1 ≤ i ≤ n are referred to as state variables for the system and the components, respectively.

In coherent systems, ξ is a function of ξ i , 1 ≤ i ≤ n:

ξ = φ(x ξ ) = φ(ξ 1 , ξ 2 , • • • , ξ n ), (19)
where

x ξ = [ξ 1 , ξ 2 , • • • , ξ n ]
is the state vector of the components. The function φ(•) in ( 19) is the structure function of the coherent system.

The state variables ξ, ξ i , 1 ≤ i ≤ n are all Boolean uncertain variables. Since ξ can be determined by ξ 1 , ξ 2 , • • • , ξ n via the structure function, ξ is a function of uncertain variables. Hence its uncertainty distribution can be obtained via the operation laws of uncertain variables [START_REF] Liu | Uncertainty theory[END_REF]. Following the operation law for Boolean uncertain variables, Liu [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF] proved the Reliability Index Theorem for coherent systems:

Theorem 1 (Reliability Index Theorem [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF]). Assume that a system contains

uncertain elements ξ 1 , ξ 2 , • • • , ξ n and has a structure function φ. If ξ 1 , ξ 2 , • • • , ξ n
are independent uncertain elements with reliability indices a 1 , a 2 , • • • , a n , respectively, then, the system reliability index a is

a =          sup φ(x1,x2,••• ,xn)=1 min 1≤i≤n ν i (x i ), if sup φ(x1,x2,••• ,xn)=1 min 1≤i≤n ν i (x i ) < 0.5 1 - sup φ(x1,x2,••• ,xn)=0 min 1≤i≤n ν i (x i ), if sup φ(x1,x2,••• ,xn)=1 min 1≤i≤n ν i (x i ) ≥ 0.5 (20)
where x i , i = 1, 2, • • • , n take value either 0 or 1, and ν i are defined by

ν i (x i ) =    a i , if x i = 1 1 -a i , if x i = 0. ( 21
)
The proof of Theorem 1 can be found in [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF].

Directly applying Theorem 1 to calculate belief reliability of a coherent system requires enumerating all possible combinations of ξ i , which is tedious and hard to apply in practice. In order to simplify the evaluation processes, we develop a system belief reliability evaluation method for coherent systems based on the concept of minimal cut sets.

Definition 3 (Minimal cut set). Suppose x = [x 1 , x 2 , • • • , x n ]
is the state vector of a coherent system whose structure function is φ. A vector x a is called a minimal cut vector if φ(x a ) = 0 and φ(x b ) = 1, ∀x b > x a . By x b > x a , we mean x b,i ≥ x a,i , 1 ≤ i ≤ n and there is at leat one i, x b,i > x a,i .

Suppose x C is a minimum cut vector. Let C(x C ) = {i : x i = 0}. Then, C(x C
) is referred to as a minimum cut set.

A minimal cut set is the smallest combination of components which will cause the systems failure if they all fail. In [START_REF] Zeng | Belief reliability: a new metrics for products' reliability[END_REF], the authors used minimal cut sets to reduce the computational costs in system belief reliability calculations.

However, their method requires a strict assumption that all the minimal cut sets are independent. In this paper, we show that the restriction is unnecessary, by proving the Minimal Cut Set Theorem, which only requires independence among the components.

Theorem 2 (Minimal Cut Set Theorem). Consider a coherent system comprising n independent components with belief reliabilities R B,i , i = 1, 2, . . . , n. If the system contains m minimal cut sets, C 1 , C 2 , . . . , C m , then, the system belief reliability is

R B,S = 1≤i≤m j∈Ci R B,j . (22)
Proof. Without loss of generality, let us assume that the ith minimal cut set C i contains n i components. Let us also assume

R B,11 ≥ R B,12 ≥ • • • R B,1j ≥ • • • ≥ R B,1n1 , R B,21 ≥ R B,22 ≥ • • • R B,2j ≥ • • • ≥ R B,2n2 , . . . R B,m1 ≥ R B,m2 ≥ • • • R B,mj ≥ • • • ≥ R B,mnm , and
R B,11 ≥ R B,21 ≥ • • • R B,j1 ≥ • • • ≥ R B,m1 ,
where R B,ij denotes the belief reliability of the jth component in the ith minimal cut set. In order to prove [START_REF] Liu | Some research problems in uncertainty theory[END_REF], we only have to prove

R B,S = R B,m1 . (23)
Equation ( 23) comes from the fact that R B,11 , R B,21 , • • • , R B,m1 are the maximum component belief reliabilities for each minimal cut set, and R B,m1 is the

minimum among R B,11 , R B,21 , • • • , R B,m1 .
The proof breaks into two cases:

1. If R B,m1 < 0.5: Since φ(x 1 , x 2 , • • • , x n ) = 1 indicates that at least one component in each minimal cut set is working, it is easy to verify that sup φ(x1,x2,••• ,xn)=1 min 1≤i≤n ν i (x i ) = min 1≤i≤m max φi(x1,x2,••• ,xn i )=1 min 1≤j≤ni ν(x ij ) (24)
where

φ i (x 1 , x 2 , • • • , x ni ) = max 1≤j≤ni x ij . Since R B,m1 ≥ R B,m2 ≥ • • • R B,mj ≥ • • • ≥ R B,mnm , we have max φm(x1,x2,••• ,xn m )=1 min 1≤j≤nm ν(x ij ) = min R B,m1 , min 2≤j≤nm (1 -R B,mj ) = R B,m1 . (25)
For

1 ≤ i ≤ m -1, if R B,i1 ≥ 0.5, from Lemma 1 in Appendix A, we have max φi(x1,x2,••• ,xn i )=1 min 1≤j≤ni ν(x ij ) ≥ 0.5 > R B,m1 ; (26)
if R B,i1 < 0.5, then, like [START_REF] Liu | Why is there a need for uncertainty theory[END_REF], we can prove that

max φi(x1,x2,••• ,xn i )=1 min 1≤j≤ni ν(x ij ) = R B,i1 ≥ R B,m1 . (27)
Substituting ( 26) and ( 27) into (24), we have

sup φi(x1,x2,••• ,xn i )=1 min 1≤j≤ni ν(x ij ) = R B,m1 < 0.5. (28)
Note that belief reliability is a reliability index. Then, from Theorem 1,

R B,S = R B,m1 . 2. If R B,m1 ≥ 0.5: Since R B,11 ≥ R B,21 ≥ • • • R B,j1 ≥ • • • ≥ R B,m1 ≥ 0.5, from Lemma 1, we have sup φ(x1,x2,••• ,xn i )=1 min 1≤j≤ni ν i (x i ) ≥ 0.5. ( 29
)
Since φ i (x 1 , x 2 , • • • , x ni ) = 0 indicates that at least in one minimal cut set, all the components fail, we have sup

φ(x1,x2,••• ,xn)=0 min 1≤i≤n ν(x i ) = max 1≤i≤m min 1≤j≤ni (1 -R B,ij ) = max 1≤i≤m (1 -R B,i1 ) = 1 -R B,m1 . (30)
Then, from Theorem 1,

R B,S = 1 - sup φ(x1,x2,••• ,xn)=0 min 1≤i≤n ν(x i ) = R B,m1 . (31)
Example 3 (Belief reliability of a series system). Consider a series system comprising n independent components with belief reliabilities R B,i , i = 1, 2, . . . , n.

It is easy to show that the system has n minimal cut sets, C 1 = {1}, C 2 = {2}, . . . , C n = {n}. Therefore, from Theorem 2, the belief reliability of the system is

R B,S = 1≤i≤n R B,i . (32)
Reference [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF] also calculates the belief reliability of a series system using the Reliability Index Theorem. The result in [START_REF] Yang | Reduction methods of type-2 uncertain variables and their applications to solid transportation problem[END_REF] is the same as that from using Theorem 1 ( [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF]). However, using Theorem 1 requires n • 2 n comparisons, while using Theorem 2 requires only n comparisons. Therefore, the computational costs can be greatly reduced by using the Minimal Cut Set Theorem.

Example 4 (Belief reliability of a parallel system). Consider a paralell system comprising n independent components with belief reliabilities R B,i , i = 1, 2, . . . , n. It is easy to show that the system has one minimal cut set, C 1 = {1, 2, . . . , n}. Therefore, from Theorem 2, the system belief reliability is

R B,S = 1≤i≤n R B,i . (33)
Reference [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF] also calculates the belief reliability of a parallel system using the Reliability Index Theorem. The result in [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF] is the same as that from using Theorem 1 ( [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF]). However, using Theorem 1 requires n • 2 n comparisons, while using Theorem 2 requires only n comparisons. Therefore, the computational costs can be greatly reduced by using the Minimal Cut Set Theorem.

Example 5 (Belief reliability of a k-out-n:G system). Consider a k-out-n:G system comprising n independent components with belief reliabilities R B,i , i = 1, 2, . . . , n. It is easy to show that the system has C (k+1) n minimal cut sets.

Each minimal cut set contains k + 1 components arbitrary chosen from the n components. Therefore, from Theorem 2, the belief reliability of the system is

R B,S = R B,k . (34)
Reference [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF] also calculates the belief reliability of a k-out-n:G system using the Reliability Index Theorem. The result in [START_REF] Zeng | Using pof models to predict system reliability considering failure collaboration[END_REF] is the same as that from using Theorem 1 ( [START_REF] Liu | Uncertain risk analysis and uncertain reliability analysis[END_REF]). However, using Theorem 1 requires n • 2 n comparisons, while using Theorem 2 requires only n comparisons. Therefore, the computational costs can be greatly reduced by using the Minimal Cut Set Theorem.

Fault tree analysis using belief reliability

In this section, we show how to calculate system belief reliability based on fault tree models. For this, we first show that Theorem 2 also applies to cut sets. A vector x CS is a cut vector if φ(x CS ) = 0. Then, CS = {i : x CS,i = 0} is defined as a cut set. All minimal cut sets are cut sets; whereas, a cut set might be necessarily be a minimal cut set since it might contain redundant elements.

If a cut set CS comprises of all the elements of a minimal cut set C and some redundant elements, C is said to be contained in CS.

Theorem 3 (Cut Set Theorem). Suppose that a coherent system has m mini-

mal cut set CS 1 , CS 2 , • • • , CS m and (l -m) cut sets CS m+1 , CS m+2 , • • • , CS l
that contain some minimal cut sets. Then, the system belief reliability can be calculated by

R B,S = 1≤i≤l j∈CSi R B,j . (35)
Proof. Let

R B,M CS = 1≤i≤m j∈CSi R B,j . (36)
Without loss of generality, let us assume that CS m+1 contains CS 1 and belief reliabilities of the redundant components are R

B,R,1 ≥ R B,R,2 ≥ R B,R,n R . Let R B,1 denote the highest belief reliability among the components in CS 1 . If R B,R,1 ≤ R B,1 , immediately we have R B,M CS = 1≤i≤m+1 j∈CSi R B,j . ( 37
) If R B,R,1 > R B,1 , (37) also holds since j∈CSm+1 R B,j = R B,R,1 > R B,1 . (38)
Similarly, we can prove that

1≤i≤l j∈CSi R B,j = R B,M CS . (39)
From Theorem 2, R B,M CS = R B,S . Hence, the theorem is proved.

The cut sets required in [START_REF] Zeng | Measuring reliability during product development considering aleatory and epistemic uncertainty[END_REF] can be enumerated from the fault tree model using the MOCUS algorithm [START_REF] Fussell | A new methodology for obtaining cut sets for fault trees[END_REF]. System belief reliability can, then, be calculated by the following algorithm:

An engineering system, the left leading edge flap (LLEF) control subsystem of the F-18 air fighters [START_REF] Dugan | A quantitative analysis of the f18 flight control system[END_REF], is used to demonstrate the developed system belief reliability analysis method. The schematic of the system is given in Fig. 2, where FCC represents flight control computer, CH represents channel, HSA represents hydraulic servo-actuator, LLEF represents left leading edge flap and RLEF represents right leading edge flap [START_REF] Dugan | A quantitative analysis of the f18 flight control system[END_REF].

The failure behavior of the system can be described by a fault tree, as shown in Fig. 3 [START_REF] Dugan | A quantitative analysis of the f18 flight control system[END_REF]. In Fig. 3, the basic events 1 -9 represent the failure of HSA-A, Algorithm 1 Belief reliability analysis based on fault tree 1: Do a depth-first-search for the logic gates in the fault tree.

2: For each logic gate, calculate the belief reliability for its output event: The belief reliability of the components can be evaluated using the procedures in [START_REF] Zeng | A new metric of reliability: Belief reliability[END_REF]. Suppose the component belief reliabilities are R Then, from [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF], the belief reliability of the LLEF control system is R B,S = R B,6 = 0.8800.

R B,out =          1≤i≤n R B,
The structure function-based method is also used to evaluate the system belief reliability. To do this, all the possible combinations of the system states need to be enumerated, which, in this case, are 2 9 = 512 states. Then, the system belief reliability is calculated based on (20). The calculated system belief reliability is R B,S = 0.8800, which is the same as the one from Algorithm 1.

According to [START_REF] Kiureghian | Aleatory or epistemic? does it matter?[END_REF], the structure function-based method requires n × 2 n = 4608 comparisons, where n is the number of components. Algorithm 1, however, requires only 10 comparisons according to [START_REF] Zio | Two methods for the structured assessment of model uncertainty by experts in performance assessments of radioactive waste repositories[END_REF]. The results demonstrate that using the developed methods can help to improve the computational efficiency of system belief reliability analysis.

Conclusion

In this paper, belief reliability was defined as an uncertain measure in uncertainty theory, due to the explicit representation of epistemic uncertainty. The Minimal Cut Set Theorem was proved, which shows how to calculate the belief reliability for coherent systems based on minimal cut sets. A system belief reliability analysis method is, then, developed based on fault tree models and applied on some numerical case studies. A comparison to the existing structure function-based method shows that the developed methods reduces the computational costs in system belief reliability analysis.

In this paper, we only consider binary systems. Many practical systems, however, are multi-state. In the future, the belief reliability evaluation method will be extended to multi-state system models. Also, the belief reliability considered in this paper is independent of time. How to model the time-dependent belief reliability is another future research direction.

Example 1 .

 1 Consider a series system of 2000 components. Suppose for each component, m d = 9 and F a = 0. It is easy to verify that both the component and the system are unlikely to fail.When using belief reliability as the reliability measure, we have to consider the effect of epistemic uncertainty, by evaluating our state of knowledge. Suppose for each component, we have F e = 3. Then, from[START_REF] De Finetti | Probability, induction, and statistics[END_REF], the belief reliability of each component is R B = 0.9987. If we regard belief reliability as a probability measure, the system belief reliability should be calculated by the product of the component belief reliabilities: R B,S = R 200 B = 0.9987 2000 = 0.074.

  ,1 = 0.9688, R B,2 = 0.9200, R B,3 = 0.9500, R B,4 = 0.9000, R B,5 = 0.8000, R B,6 = 0.8800, R B,7 = 0.9600, R B,8 = 0.9700, R B,9 = 0.9500, respectively. From Algorithm 1, the belief reliability of the system isR B,S = R B,1 ∧ R B,2 ∧ R B,3 ∧ (41) (R B,5 ∧ R B,8 ) ∨ (R B,6 ∧ R B,9 ) ∧ (R B,4 ∨ R B,5 ∨ R B,6 ∨ R B,7 )

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Epistemic uncertainty effect on the aleatory distribution of the performance margin (Adapted from [37])

  Definition 2 (Mathematical definition of belief reliability). Let the universal set Γ = {γ 1 , γ 2 }, where γ 1 represents the working state of a system or component, while γ 2 represents the failure state. Then, belief reliability R B is defined as the uncertain measure of the event Λ 1 = {γ 1 },

	capable to compensate for the extra dispersion induced by epistemic uncertainty.
	Compared to possibility theory, uncertainty theory follows the Duality Axiom,
	which prevents the counter-intuitive examples such as that in Example 2. Hence,
	belief reliability is assumed to be an uncertain measure in this paper.
	Examples 1 and 2, we can see that to model belief reliability, we need a
	mathematical theory which can compensate the conservatism in the individual
	belief degree and satisfy the duality axiom. Compared to probability theory,
	uncertainty theory differs in the Product Axiom, where a minimum operator is
	used instead of the product operator, indicating that the uncertainty theory is

  in,i , for an OR gate, : R B,S ← R B,out,T E , where T E represents top event.

	(40)
	R B,in,i , for an AND gate,
	1≤i≤n
	left asymmetry control unit, LLEF, CH1, CH2, CH3, CH4, FCC-A and FCC-B,
	respectively.

34: return R B,S .

Acknowledgment This work has been performed within the initiative of the Center for Resilience and Safety of Critical Infrastructures (CRESCI, http://cresci.cn). The research of Prof. Rui Kang is supported by National Natural Science Foundation of China (NSFC) [grant number 61573043]. The research of Dr. Meilin Wen is supported by National Natural Science Foundation of China (NSFC) [grant number 71671009].

If the structure function of the system φ is:

and there is at least one R B,i such that R B,i ≥ 0.5, then we have

Proof. The proof breaks into two cases:

Without loss of generality, we assume that there exists a k, k

such that R B,k ≥ 0.5. Since R n < 0.5, there exists a j ∈ (k, n), where

where