A reduced sampling approach for reliability verification

Reliability demonstration tests frequently require large sample sizes which are objectionable from both a cost and time standpoint. The approach presented, referred to as bias sampling, offers a means to demonstrate conformance to reliability objectives with reduced test samples and test time. The reduction is achieved by selection and test of samples from the lower-median sub-population of a random sample. The prediction assumes that the entire population of samples was tested. Rationale for the approach is presented along with guidelines for sample selection. Two application examples illustrate how the methodology is being employed at Chrysler Corporation.     

An efficient analytical Bayesian method for reliability and system response updating based on Laplace and inverse first-order reliability computations

This paper presents an efficient analytical Bayesian method for reliability and system response updating without using simulations. The method includes additional information such as measurement data via Bayesian modeling to reduce estimation uncertainties. Laplace approximation method is used to evaluate Bayesian posterior distributions analytically. An efficient algorithm based on inverse first-order reliability method is developed to evaluate system responses given a reliability index or confidence interval. Since the proposed method involves no simulations such as Monte Carlo or Markov chain Monte Carlo simulations, the overall computational efficiency improves significantly, particularly for problems with complicated performance functions. A practical fatigue crack propagation problem with experimental data, and a structural scale example are presented for methodology demonstration. The accuracy and computational efficiency of the proposed method are compared with traditional simulation-based methods.     

A Bayesian perspective on age replacement with minimal repair

In this article, a Bayesian approach is developed for determining an optimal age replacement policy with minimal repair. By incorporating minimal repair, planned replacement, and unplanned replacement, the mathematical formulas of the expected cost per unit time are obtained for two cases – the infinite-horizon case and the one-replacement-cycle case. For each case, we show that there exists a unique and finite optimal age for replacement under some reasonable conditions. When the failure density is Weibull with uncertain parameters, a Bayesian approach is established to formally express and update the uncertain parameters for determining an optimal age replacement policy. Further, various special cases are discussed in detail. Finally, a numerical example is given.     

Bayesian decision analysis and reliability certification

Reliability certification is set as a problem of Bayesian Decision Analysis. Uncertainties about the system reliability are quantified by assuming the parameters of the models describing the stochastic behavior of components as random variables. A utility function quantifies the relative value of each possible level of system reliability after having been accepted or the opportunity loss of the same level if the system has been rejected. A decision about accepting or rejecting the system can be made either on the basis of the existing prior assessment of uncertainties or after obtaining further information through testing of the components or the system at a cost. The concepts of value of perfect information, expected value of sample information and the expected net gain of sampling are specialized to the reliability of a multicomponent system to determine the optimum component testing scheme prior to deciding on the system's certification. A component importance ranking is proposed on the basis of the expected value of perfect information about the reliability of each component. The proposed approach is demonstrated on a single component system failing according to a Poisson random process and with natural conjugate prior probability density functions (pdf) for the failure rate and for a multicomponent system under general assumptions.     

A discrete-time Bayesian network reliability modeling and analysis framework

Dependability tools are becoming an indispensable tool for modeling and analyzing (critical) systems. However the growing complexity of such systems calls for increasing sophistication of these tools. Dependability tools need to not only capture the complex dynamic behavior of the system components, but they must be also easy to use, intuitive, and computationally efficient. In general, current tools have a number of shortcomings including lack of modeling power, incapacity to efficiently handle general component failure distributions, and ineffectiveness in solving large models that exhibit complex dependencies between their components. We propose a novel reliability modeling and analysis framework based on the Bayesian network (BN) formalism. The overall approach is to investigate timed Bayesian networks and to find a suitable reliability framework for dynamic systems. We have applied our methodology to two example systems and preliminary results are promising. We have defined a discrete-time BN reliability formalism and demonstrated its capabilities from a modeling and analysis point of view. This research shows that a BN based reliability formalism is a powerful potential solution to modeling and analyzing various kinds of system components behaviors and interactions. Moreover, being based on the BN formalism, the framework is easy to use and intuitive for non-experts, and provides a basis for more advanced and useful analyses such as system diagnosis.     

Component reliability assessment and assurance using a new step-stress test

Several features of accelerated reliability testing are surveyed in this paper. A new technique for the practical evaluation of reliability under thermally induced acceleration is introduced. This procedure is termed the reverse-step-stress test, and involves a decrease in applied stress after some pre-specified failure criterion has been met within a fixed sample of components. An analysis is presented which confirms that the reverse-step-stress test can reveal failures having low activation energies that would not be detectable in conventional constant-stress, high temperature testing.     

Design of PH-based accelerated life testing plans under multiple-stress-type

Accelerated life testing (ALT) is used to obtain failure time data quickly under high stress levels in order to predict product life performance under design stress conditions. Most of the previous work on designing ALT plans is focused on the application of a single stress. However, as components or products become more reliable due to technological advances, it becomes more difficult to obtain significant amount of failure data within reasonable amount of time using single stress only. Multiple-stress-type ALTs have been employed as a means of overcoming such difficulties. In this paper, we design optimum multiple-stress-type ALT plans based on the proportional hazards model. The optimum combinations of stresses and their levels are determined such that the variance of the reliability estimate of the product over a specified period of time is minimized. The use of the model is illustrated using numerical example, and sensitivity analysis shows that the resultant optimum ALT plan is robust to the deviation in model parameters.     

A two-stage sampling plan for bogey tests

When accelerated life tests can be applied to simulate the normal product operating conditions, engineers usually terminate the test upon successfully running to a multiple of a prespecified bogey without any failure to demonstrate a required minimum reliability level. This testing philosophy is called ‘bogey test’ (or extended test) in the automotive industry and is a subset of type I censored tests in standard reliability literature. Sometimes engineers encounter the difficulty of using this reliability demonstration approach when incidents do occur during the test. The incident might be a legitimate failure or a withdrawal caused by external forces such as a broken fixture of a non-functional power supply. This paper derives a two-stage sampling plan for possible back-up solution on planning and running a bogey test with possible occurrence of incidents. A Weibull distribution with a given shape parameter is assumed for the underlying life characteristic.     

Bayesian updating of reliability of civil infrastructure facilities based on condition-state data and fault-tree model

This paper considers a difficult but practical circumstance of civil infrastructure management—deterioration/failure data of the infrastructure system are absent while only condition-state data of its components are available. The goal is to develop a framework for estimating time-varying reliabilities of civil infrastructure facilities under such a circumstance. A novel method of analyzing time-varying condition-state data that only reports operational/non-operational status of the components is proposed to update the reliabilities of civil infrastructure facilities. The proposed method assumes that the degradation arrivals can be modeled as a Poisson process with unknown time-varying arrival rate and damage impact and that the target system can be represented as a fault-tree model. To accommodate large uncertainties, a Bayesian algorithm is proposed, and the reliability of the infrastructure system can be quickly updated based on the condition-state data. Use of the new method is demonstrated with a real-world example of hydraulic spillway gate system.     

Estimation of mixed Weibull distribution parameters using the SCEM-UA algorithm: Application and comparison with MLE in automotive reliability analysis

The shuffled complex-evolution metropolis algorithm (SCEM-UA) is used to estimate mixed Weibull distribution parameters in automotive reliability analysis. The results are compared with maximum likelihood estimation (MLE) results. The comparison shows that, in the examples given, SCEM-UA can deliver more accurate results than MLE overall.     

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.     

System reliability and sensitivity factors via the MPPSS method

A procedure is presented for obtaining the reliability of series, parallel, and mixed systems. The method, called Most Probable Point System Simulation (MPPSS) is simple, and numerically-based. We do not make any claims of analytic advance. Nevertheless, the procedure is more accurate than current analytically-based bounding methods (e.g. bi-modal) and it is computationally efficient. In addition, the method can be used to obtain system sensitivity factors, that is, the importance of each random variable to the system reliability. The procedure is demonstrated through three example illustrations.     

Modelling the reliability of search and rescue operations with Bayesian Belief Networks

This paper uses a Bayesian Belief Networks (BBN) methodology to model the reliability of Search And Rescue (SAR) operations within UK Coastguard (Maritime Rescue) coordination centres. This is an extension of earlier work, which investigated the rationale of the government's decision to close a number of coordination centres. The previous study made use of secondary data sources and employed a binary logistic regression methodology to support the analysis. This study focused on the collection of primary data through a structured elicitation process, which resulted in the construction of a BBN. The main findings of the study are that statistical analysis of secondary data can be used to complement BBNs. The former provided a more objective assessment of associations between variables, but was restricted in the level of detail that could be explicitly expressed within the model due to a lack of available data. The latter method provided a much more detailed model, but the validity of the numeric assessments was more questionable. Each method can be used to inform and defend the development of the other. The paper describes in detail the elicitation process employed to construct the BBN and reflects on the potential for bias.     

Comparison of evidence theory and Bayesian theory for uncertainty modeling

This paper compares Evidence Theory (ET) and Bayesian Theory (BT) for uncertainty modeling and decision under uncertainty, when the evidence about uncertainty is imprecise. The basic concepts of ET and BT are introduced and the ways these theories model uncertainties, propagate them through systems and assess the safety of these systems are presented. ET and BT approaches are demonstrated and compared on challenge problems involving an algebraic function whose input variables are uncertain. The evidence about the input variables consists of intervals provided by experts. It is recommended that a decision-maker compute both the Bayesian probabilities of the outcomes of alternative actions and their plausibility and belief measures when evidence about uncertainty is imprecise, because this helps assess the importance of imprecision and the value of additional information. Finally, the paper presents and demonstrates a method for testing approaches for decision under uncertainty in terms of their effectiveness in making decisions.     

Satellite and satellite subsystems reliability: Statistical data analysis and modeling

Reliability has long been recognized as a critical attribute for space systems. Unfortunately, limited on-orbit failure data and statistical analyses of satellite reliability exist in the literature. To fill this gap, we recently conducted a nonparametric analysis of satellite reliability for 1584 Earth-orbiting satellites launched between January 1990 and October 2008. In this paper, we extend our statistical analysis of satellite reliability and investigate satellite subsystems reliability. Because our dataset is censored, we make extensive use of the Kaplan–Meier estimator for calculating the reliability functions. We derive confidence intervals for the nonparametric reliability results for each subsystem and conduct parametric fits with Weibull distributions using the maximum likelihood estimation (MLE) approach. We finally conduct a comparative analysis of subsystems failure, identifying the “culprit subsystems” that drive satellite unreliability. The results here presented should prove particularly useful to the space industry for example in redesigning subsystem test and screening programs, or providing an empirical basis for redundancy allocation.     

Statistical analysis and reliability prediction with short fatigue crack data

This paper proposes a probability model to describe the growth of short fatigue cracks. The model defines the length of each crack in a specimen as a random quantity, which is a function of randomly varying local properties of the material microstructure. Once the model has been described, the paper addresses two questions: first, statistical inference, i.e. the fitting of the model parameters to data on crack lengths; and secondly, predicting the future behaviour of observed cracks or cracks in a new specimen. By defining failure of a specimen to be the time at which the largest crack exceeds a certain length, the solution to the prediction problem can be used to calculate a probability that the specimen has failed at any future time. The probability model for crack lengths is called a population model, and the statistical inference uses the ideas of Bayesian statistics. Both these concepts are described. With a population model, the solution to statistical inference and prediction requires quite complicated Monte Carlo simulation techniques, which are also described.     

Bayesian Approach to Assessing Uncertainty and Calculating a Reference Value in Key Comparison Experiments

International experiments called Key Comparisons pose an interesting statistical problem, the estimation of a quantity called a Reference Value. There are many possible forms that this estimator can take. Recently, this topic has received much international attention. In this paper, it is argued that a fully Bayesian approach to this problem is compatible with the current practice of metrology, and can easily be used to create statistical models which satisfy the varied properties and assumptions of these experiments.