Reliability assessment for systems suffering common cause failure based on Bayesian networks and proportional hazards model

The Bayesian network (BN) is an efficient tool for probabilistic modeling and causal inference, and it has gained considerable attentions in the field of reliability assessment. The common cause failure (CCF) is simultaneous failure of multiple elements in a system under a common cause, and it is a common phenomenon in engineering systems with dependent elements. Several models and methods have been proposed for modeling and assessment of complex systems with CCF. In this paper, a new reliability assessment method is proposed for the systems suffering from CCF in a dynamic environment. The CCF among components is characterized by a BN, which allows for bidirectional reasoning. A proportional hazards model is applied to capture the dynamic working environment of components and then the reliability function of the system is obtained. The proposed method is validated through an illustrative example, and some comparative studies are also presented.     

Analysis of correlated multivariate degradation data in accelerated reliability growth

Modern engineering systems have become increasingly complex and at the same time are expected to be developed faster. To shorten the product development time, organizations commonly conduct accelerated testing on a small number of units to help identify failure modes and assess reliability. Many times design changes are made to mitigate or reduce the likelihood of such failure modes. Since failure-time data are often scarce in reliability growth programs, existing statistical approaches used for predicting the reliability of a system about to enter the field are faced with significant challenges. In this work, a statistical model is proposed to utilize degradation data for system reliability prediction in an accelerated reliability growth program. The model allows the components in the system to have multiple failure modes, each associated with a monotone stochastic degradation process. To take into account unit-to-unit variation, the random effects of degradation parameters are explicitly modeled. Moreover, a mean-degradation-stress relationship is introduced to quantify the effects of different accelerating variables on the degradation processes, and a copula function is utilized to model the dependency among different degradation processes. Both a maximum likelihood (ML) procedure and a Bayesian alternative are developed for parameter estimation in a two-stage process. A numerical study illustrates the use of the proposed model and identifies the cases where the Bayesian method is preferred and where it is better to use the ML alternative.     

Reliability assessment of complex electromechanical systems: A network perspective

The network theory is widely applied to improve the reliability of a complex electromechanical system. In this application, system reliability assessment with network theory has been paid a great deal of attention. Because of instrument malfunctions, staff omissions, imperfect inspection strategies, and complex structures, field failure data are often subject to interval censoring, making the holistic reliability assessment becomes a difficult task. Most traditional methods assume reliability of critical components or partial reliability as system reliability, which may cause a large bias in system reliability estimation. This paper proposes a novel method to evaluate and predict the system reliability of a complex electromechanical system subject to the insufficient fault data problem from a network perspective. First, the system modeling based on network theory is developed to describe the topology of a holistic system. Second, interval‐valued intuitionistic hesitant fuzzy number is proposed in order to solve insufficient data for single component. Then, a new measure—comprehensive reliability—that can reflect the reliability of nodes in combination with functional properties and topological properties, which are formulated by failure data and network model, respectively, is constructed for system reliability assessment. Subsequently, an improved system reliability model based on percolation theory is given in terms of comprehensive reliability of nodes. Finally, to verify the effectiveness of the proposed method, a simulation and a real case study for traction system are implemented.     

A system reliability estimation method by the fourth moment saddle point approximation and copula functions

Despite many advances in the field of computational system reliability analysis, estimating the joint probability distribution of correlated non-normal state variables on the basis of incomplete statistical data brings great challenges for engineers. To avoid multidimensional integration, system reliability estimation usually requires the calculation of marginal failure probability and joint failure probability. The current article proposed an integrated approach for estimating system reliability on the basis of the high moment method, saddle point approximation, and copulas. First, the statistic moment estimation based on the stochastic perturbation theory is presented. Thereafter, by constructing CGF (concise cumulant generating function) for the state variable with its first four statistical moments, a fourth moment saddle point approximation method is established for the component reliability estimation. Second, the copula theory is briefly introduced and extensively utilized two-dimensional copulas are presented. The best fit copula for estimating the probability of system failure is selected according to the AIC (Akaike Information Criterion). Finally, the derived method is applied to three numerical examples for the sake of a comprehensive validation.     

Software reliability analysis for safety-critical and control systems

The transition from analog to digital safety-critical instrumentation and control (I&C) systems has introduced new challenges for software experts to deliver increased software reliability. Since the 1970s, researchers are continuing to propose software reliability models for reliability estimation of software. However, these approaches rely on the failure history for the assessment of reliability. Due to insufficient failure data, these models fail to predict the reliability of safety critical systems. This paper utilizes the Bayesian update methodology and proposes a framework for the reliability assessment of the safety-critical systems (SCSs). The proposed methodology is validated using experiments performed on real data of 12 safety-critical control systems of nuclear power plants.     

Application of Bayesian Methods for Age‐Dependent Reliability Analysis

In this article, the authors present a general methodology for age‐dependent reliability analysis of degrading or ageing components, structures and systems. The methodology is based on Bayesian methods and inference—its ability to incorporate prior information and on ideas that ageing can be thought of as age‐dependent change of beliefs about reliability parameters (mainly failure rate), when change of belief occurs not only because new failure data or other information becomes available with time but also because it continuously changes due to the flow of time and the evolution of beliefs. The main objective of this article is to present a clear way of how practitioners can apply Bayesian methods to deal with risk and reliability analysis considering ageing phenomena. The methodology describes step‐by‐step failure rate analysis of ageing components: from the Bayesian model building to its verification and generalization with Bayesian model averaging, which as the authors suggest in this article, could serve as an alternative for various goodness‐of‐fit assessment tools and as a universal tool to cope with various sources of uncertainty. The proposed methodology is able to deal with sparse and rare failure events, as is the case in electrical components, piping systems and various other systems with high reliability. In a case study of electrical instrumentation and control components, the proposed methodology was applied to analyse age‐dependent failure rates together with the treatment of uncertainty due to age‐dependent model selection. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of discrete‐time Bayesian network on reliability analysis of uncertain system with common cause failure

With the increasing complexity of engineering systems, reliability analysis and evaluation of systems with traditional methods can't meet practical engineering requirements. Based on limited experimental conditions, lack of data, complex structure models, insufficient cognitive abilities, and many other issues, people have to consider many uncertain factors in system reliability research. Besides, common cause failure (CCF) has become an important factor of system failure. In this paper, a discrete‐time Bayesian network (DTBN) associated with an eight‐rotor unmanned aerial vehicle (UAV) system is presented to discuss above problems. In this approach, the system is assumed as a two‐state system. After that, interval analysis theory is employed to deal with uncertainty. We consider the four sets of auxiliary propellers in the auxiliary power group as a 3/8 voting system, and β factor model is used to process CCF in the auxiliary power group. The proposed methods prove the validity of proposing interval analysis theory to solve uncertain problems and it is necessary to consider reducing or avoiding CCFs in system.     

System Reliability and Component Importance Under Dependence: A Copula Approach

System reliability and component importance are of great interest in reliability modeling, especially when the components within the system are dependent. We characterize the influence of dependence structures on system reliability and component importance in coherent systems with discrete marginal distributions. The effects of dependence are captured through copula theory. We extend our framework to coherent multi-state system. Applications of the derived results are demonstrated using a Gaussian copula, which yields simple interpretations. Simulations and two examples are presented to demonstrate the importance of modeling dependence when estimating system reliability and ranking of component importance. Proofs, algorithms, code, and data are provided in supplementary materials available online.     

A Bayesian framework for accelerated reliability growth testing with multiple sources of uncertainty

Reliability growth tests are often used for achieving a target reliability for complex systems via multiple test‐fix stages with limited testing resources. Such tests can be sped up via accelerated life testing (ALT) where test units are exposed to harsher‐than‐normal conditions. In this paper, a Bayesian framework is proposed to analyze ALT data in reliability growth. In particular, a complex system with components that have multiple competing failure modes is considered, and the time to failure of each failure mode is assumed to follow a Weibull distribution. We also assume that the accelerated condition has a fixed time scaling effect on each of the failure modes. In addition, a corrective action with fixed ineffectiveness can be performed at the end of each stage to reduce the occurrence of each failure mode. Under the Bayesian framework, a general model is developed to handle uncertainty on all model parameters, and several special cases with some parameters being known are also studied. A simulation study is conducted to assess the performance of the proposed models in estimating the final reliability of the system and to study the effects of unbiased and biased prior knowledge on the system‐level reliability estimates.     

Sensibility of Bayesian inference methods for reliability prediction of ageing systems,case of Diesel locomotives

This paper addresses the problem of reliability analysis of in-service identical systems when a limited number of lifetime data is available compared to censored ones. Lifetime (resp. censored) data characterise the life of failed (resp. non-failed) systems in the sample. Because, censored data induce biassed estimators of reliability model parameters, a methodology approach is proposed to overcome this inconvenience and improve the accuracy of the parameter estimation based on Bayesian inference methods. These methods combine, in an effective way, system’s life data and expert opinions learned from failure diagnosis of similar systems. Three Bayesian inference methods are considered: Classical Bayesian, Extended Bayesian and Bayesian Restoration Maximisation methods. Given a sample of lifetime data, simulated according to prior opinions of maintenance expert, a sensibility analysis of each Bayesian method is performed. Reliability analysis of critical subsystems of Diesel locomotives is established under the proposed methodology approach. The relevance of each Bayesian inference methods with respect to collected reliability data of critical subsystems and expert opinions is discussed.     

Data-driven reliability assessment with scarce samples considering multidimensional dependence

This study proposes a data-driven method for assessing reliability, based on the scarce input dataset with multidimensional correlation. Since considering the distribution parameters estimated from the scarce dataset as those of the population may lead to epistemic uncertainty, the bootstrap resampling algorithm is adopted to infer the distribution parameters as interval parameters. To account for the variable dependence, vine copula theory is utilized to construct the joint probability density function (PDF) of input variables, and maximum likelihood estimation (MLE) and Akaike information criterion (AIC) analysis are employed to select optimal copulas based on the samples for the vine structure. Subsequently, the failure probability bounds of a response function are calculated based on the constructed joint PDF with interval distribution parameters by the active learning Kriging (AK) method combining the sparse grid integration (SGI) method. Finally, several examples are provided to demonstrate the feasibility and efficiency of the proposed method.     

基于Copula函数的并联结构系统可靠度分析

不完备概率信息条件下变量联合分布函数的确定及其对结构系统可靠度的影响还缺少系统地研究,该文目的在于研究表征变量间相关性的Copula函数对结构系统可靠度的影响规律。首先,简要介绍了变量联合分布函数构造的Copula函数方法。其次,提出了并联系统失效概率计算方法,并推导了相应的计算公式。最后以几种典型Copula函数为例研究了Copula函数类型对结构并联系统可靠度的影响规律。结果表明:表征变量间相关性的Copula函数类型对结构系统可靠度具有明显的影响,不同Copula函数计算的系统失效概率存在明显的差别,这种差别随构件失效概率的减小而增大。当并联系统的失效区域位于Copula函数尾部时,Copula函数的尾部相关性对系统可靠度有明显的影响,计算的失效概率比没有尾部相关性的Copula函数的失效概率大。当组成并联系统的两构件功能函数间正相关时,系统失效概率随相关系数的增大而增加;当构件功能函数间负相关时,系统失效概率随相关系数的增大而减小。此外,无论构件失效概率和变量间相关系数如何变化,Copula函数计算的失效概率都位于系统失效概率的上下限内。     

Integration of computation and testing for reliability estimation

This paper develops a methodology to integrate reliability testing and computational reliability analysis for product development. The presence of information uncertainty such as statistical uncertainty and modeling error is incorporated. The integration of testing and computation leads to a more cost-efficient estimation of failure probability and life distribution than the tests-only approach currently followed by the industry. A Bayesian procedure is proposed to quantify the modeling uncertainty using random parameters, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. An adaptive method is developed to determine the number of tests needed to achieve a desired confidence level in the reliability estimates, by combining prior computational prediction and test data. Two kinds of tests — failure probability estimation and life estimation — are considered. The prior distribution and confidence interval of failure probability in both cases are estimated using computational reliability methods, and are updated using the results of tests performed during the product development phase.     

General network reliability problem and its efficient solution by Subset Simulation

Complex technological networks designed for distribution of some resource or commodity are a pervasive feature of modern society. Moreover, the dependence of our society on modern technological networks constantly grows. As a result, there is an increasing demand for these networks to be highly reliable in delivering their service. As a consequence, there is a pressing need for efficient computational methods that can quantitatively assess the reliability of technological networks to enhance their design and operation in the presence of uncertainty in their future demand, supply and capacity. In this paper, we propose a stochastic framework for quantitative assessment of the reliability of network service, formulate a general network reliability problem within this framework, and then show how to calculate the service reliability using Subset Simulation, an efficient Markov chain Monte Carlo method that was originally developed for estimating small failure probabilities of complex dynamic systems. The efficiency of the method is demonstrated with an illustrative example where two small-world network generation models are compared in terms of the maximum-flow reliability of the networks that they produce.     

Predictive maintenance of complex system with multi-level reliability structure

Onboard sensors, which constantly monitor the states of a system and its components, have made the predictive maintenance (PdM) of a complex system possible. To date, system reliability has been extensively studied with the assumption that systems are either single-component systems or they have a deterministic reliability structure. However, in many realistic problems, there are complex multi-component systems with uncertainties in the system reliability structure. This paper presents a PdM scheme for complex systems by employing discrete time Markov chain models for modelling multiple degradation processes of components and a Bayesian network (BN) model for predicting system reliability. The proposed method can be considered as a special type of dynamic Bayesian network because the same BN is repeatedly used over time for evaluating system reliability and the inter-time–slice connection of the same node is monitored by a sensor. This PdM scheme is able to make probabilistic inference at any system level, so PdM can be scheduled accordingly.     

A case study for quantifying system reliability and uncertainty

The ability to estimate system reliability with an appropriate measure of associated uncertainty is important for understanding its expected performance over time. Frequently, obtaining full-system data is prohibitively expensive, impractical, or not permissible. Hence, methodology which allows for the combination of different types of data at the component or subsystem levels can allow for improved estimation at the system level. We apply methodologies for aggregating uncertainty from component-level data to estimate system reliability and quantify its overall uncertainty. This paper provides a proof-of-concept that uncertainty quantification methods using Bayesian methodology can be constructed and applied to system reliability problems for a system with both series and parallel structures.     

A Copula Approach in the Point Estimate Method for Reliability Engineering

In order to arrive at realistic results in statistical analysis, it is often advisable to consider involved uncertainties as random variables. An important aspect in this context is the evaluation of the importance of parameter uncertainty. Because of the complexity of computational models, the point estimate method is usually adopted as an easy‐run approach for approximating the statistical moments of a system/model in a reliability analysis. The efficiency of this method highly depends on the correlation coefficient. However, the complex nature of parameters in computational problems often exhibits a nonlinear relationship. This paper aims to develop an original and efficient point estimate method based on the copula approach for reliability engineering problems. The paper discusses the use of the copula theory in the point estimate method for computing the statistical moments of a function involving random variables. The study performs two engineering applications to demonstrate the benefits of this approach. The performance of this proposed method can significantly improve the quality of the results in using the point estimate method when a nonlinear relationship exists between the parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Copula‐function‐based analysis model and dynamic reliability of a gear transmission system considering failure correlations

In this study, an improved prediction model is introduced to assess the residual strength of gear transmission systems. The residual strength and stress‐strength interference theories are then combined to facilitate the study of the reliability of the gears and bearings in these systems. Based on the copula function, the correlation between the gear contact and bending failure is determined. The relationship between the copula function and reliability is used to determine the dynamic reliability of the gears while considering multiple correlated failure modes. In this manner, more accurate and dynamic reliability attenuation characteristics are obtained for the gear transmission system.     

Modelling dependable systems using hybrid Bayesian networks

A hybrid Bayesian network (BN) is one that incorporates both discrete and continuous nodes. In our extensive applications of BNs for system dependability assessment, the models are invariably hybrid and the need for efficient and accurate computation is paramount. We apply a new iterative algorithm that efficiently combines dynamic discretisation with robust propagation algorithms on junction tree structures to perform inference in hybrid BNs. We illustrate its use in the field of dependability with two example of reliability estimation. Firstly we estimate the reliability of a simple single system and next we implement a hierarchical Bayesian model. In the hierarchical model we compute the reliability of two unknown subsystems from data collected on historically similar subsystems and then input the result into a reliability block model to compute system level reliability. We conclude that dynamic discretisation can be used as an alternative to analytical or Monte Carlo methods with high precision and can be applied to a wide range of dependability problems.