Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley & Sons, Ltd.     

An additive Chen-Weibull distribution and its applications in reliability modeling

In this article, we introduce a new lifetime distribution with increasing and bathtub-shaped failure rates. Some statistical properties of the proposed distribution are studied. We use the method of maximum likelihood for estimating the model parameters and reliability characteristics and discuss the interval estimates using asymptotic confidence intervals and bootstrap confidence intervals on one hand, and we provide Bayes estimators and highest posterior density intervals for the parameters via Hamiltonian Monte Carlo simulation method on the other hand. We demonstrate the superiority of the proposed distribution by fitting two reliability data sets well-known from references.     

Reliability inference for VGA adapter from dual suppliers based on contaminated type‐I interval‐censored data

Type‐I interval‐censoring scheme only documents the number of failed units within two prespecified consecutive exam times at the larger time point after putting all units on test at the initial time schedule. It is challenging to use the collected information from type‐I interval‐censoring scheme to evaluate the reliability of unit when not all admitted units are operated or tested at the same initial time and a majority of units are randomly selected to replace the failed test units at unrecorded time points. Moreover, the lifetime distribution of all pooled units from dual resources usually follows a mixture distribution. To overcome these two problems, a two‐stage inference process that consists of a data‐cleaning step and a parameter estimation step via either Markov chain Monte Carlo (MCMC) algorithm or profile likelihood method is proposed based on the contaminated type‐I interval‐censored sample from a mixture distribution with unknown proportion. An extensive simulation study is conducted under the mixture smallest extreme value distributions to evaluate the performance of the proposed method for a case study. Finally, the proposed methods are applied to the mixture lifetime distribution modeling of video graphics array adapters for the support of reliability decision.     

More on the mis‐specification of the shape parameter with Weibull‐to‐exponential transformation

When lifetimes follow Weibull distribution with known shape parameter, a simple power transformation could be used to transform the data to the case of exponential distribution, which is much easier to analyze. Usually, the shape parameter cannot be known exactly and it is important to investigate the effect of mis‐specification of this parameter. In a recent article, it was suggested that the Weibull‐to‐exponential transformation approach should not be used as the confidence interval for the scale parameter has very poor statistical property. However, it would be of interest to study the use of Weibull‐to‐exponential transformation when the mean time to failure or reliability is to be estimated, which is a more common question. In this paper, the effect of mis‐specification of Weibull shape parameters on these quantities is investigated. For reliability‐related quantities such as mean time to failure, percentile lifetime and mission reliability, the Weibull‐to‐exponential transformation approach is generally acceptable. For the cases when the data are highly censored or when small tail probability is concerned, further studies are needed, but these are known to be difficult statistical problems for which there are no standard solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Second‐order reliability method‐based inverse reliability analysis using Hessian update for accurate and efficient reliability‐based design optimization

First‐order reliability method (FORM) has been mostly utilized for solving reliability‐based design optimization (RBDO) problems efficiently. However, second‐order reliability method (SORM) is required in order to estimate a probability of failure accurately in highly nonlinear performance functions. Despite accuracy of SORM, its application to RBDO is quite challenging due to unaffordable numerical burden incurred by a Hessian calculation. For reducing the numerical efforts, a quasi‐Newton approach to approximate the Hessian is introduced in this study instead of calculating the true Hessian. The proposed SORM with the approximated Hessian requires computations only used in FORM, leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes a generalized chi‐squared distribution in order to achieve better accuracy. Furthermore, SORM‐based inverse reliability method is proposed in this study. An accurate reliability index corresponding to a target probability of failure is updated using the proposed SORM. Two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed. The proposed SORM‐based inverse analysis is then extended to RBDO in order to obtain a reliability‐based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system. The numerical study results show that the proposed reliability analysis and RBDO achieve efficiency of FORM and accuracy of SORM at the same time. Copyright © 2014 John Wiley & Sons, Ltd.     

A Bayesian zero‐failure reliability demonstration test of high quality electro‐explosive devices

Usually, for high reliability products the production cost is high and the lifetime is much longer, which may not be observable within a limited time. In this paper, an accelerated experiment is employed in which the lifetime follows an exponential distribution with the failure rate being related to the accelerated factor exponentially. The underlying parameters are also assumed to have the exponential prior distributions. A Bayesian zero‐failure reliability demonstration test is conducted to design forehand the minimum sample size and testing length subject to a certain specified reliability criterion. Probability of passing the test design as well as predictive probability for additional experiments is also derived. Sensitivity analysis of the design is investigated by a simulation study. Copyright © 2009 John Wiley & Sons, Ltd.     

Two‐Stage Reliability Sampling Plans with Bogey Tests

This article considers the design of two‐stage reliability test plans. In the first stage, a bogey test was performed, which will allow the user to demonstrate reliability at a high confidence level. If the lots pass the bogey test, the reliability sampling test is applied to the lots in the second stage. The purpose of the proposed sampling plan was to test the mean time to failure of the product as well as the minimum reliability at bogey. Under the assumption that the lifetime distribution follows Weibull distribution and the shape parameter is known, the two‐stage reliability sampling plans with bogey tests are developed and the tables for users are constructed. An illustrative example is given, and the effects of errors in estimates of a Weibull shape parameter are investigated. A comparison of the proposed two‐stage test with corresponding bogey and one‐stage tests was also performed. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of Field Reliability Based on Aggregate Lifetime Data

Because of the exponential distribution assumption, many reliability databases recorded data in an aggregate way. Instead of individual failure times, each aggregate data point is a summation of a series of collective failures representing the cumulative operating time of one component position from system commencement to the last component replacement. The data format is different from traditional lifetime data and the statistical inference is challenging. We first model the individual component lifetime by a gamma distribution. Confidence intervals for the gamma shape parameter can be constructed using a scaled χ² approximation to a modified ratio of the geometric mean to the arithmetic mean, while confidence intervals for the gamma rate and mean parameters, as well as quantiles, are obtained using the generalized pivotal quantity method. We then fit the data using the inverse Gaussian (IG) distribution, a useful lifetime model for failures caused by degradation. Procedures for point estimation and interval estimation of parameters are developed. We also propose an interval estimation method for the quantiles of an IG distribution based on the generalized pivotal quantity method. An illustrative example demonstrates the proposed inference methods. Supplementary materials for this article are available online.     

Estimation of time‐to‐failure distribution derived from a degradation model using fuzzy clustering

Some life tests are terminated with few or no failures. In such cases, a recent approach is to obtain degradation measurements of product performance that may contain some useful information about product reliability. Generally degradation paths of products are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the time‐to‐failure distribution can be estimated. In some cases, the patterns of a few degradation paths are different from those of most degradation paths in a test. Therefore, this study develops a weighted method based on fuzzy clustering procedure to robust estimation of the underlying parameters and time‐to‐failure distribution. The method will be studied on a real data set. Copyright © 2000 John Wiley & Sons, Ltd.     

Field Failure Data: an Alternative Proposal for Reliability Estimation

Product reliability is a very important issue for the competitive strategy of industries. In order to estimate a product's reliability, parametric inferential methods are required to evaluate survival test data, which happens to be a fairly expensive data source. Such costly information usually imposes additional compromises in the product development and new challenges to be overcome throughout the product's life cycle. However, manufacturers also keep field failure data for warranty and maintenance purposes, which can be a low‐cost data source for reliability estimation. Field‐failure data are very difficult to evaluate using parametric inferential methods due to their small and highly censored samples, quite often representing mixed modes of failure. In this paper a method for reliability estimation using field failure data is proposed. The proposal is based on the use of non‐parametric inferential methods, associated with resampling techniques to derive confidence intervals for the reliability estimates. Test results show the adequacy of the proposed method to calculate reliability estimates and their confidence interval for different populations, including cases with highly right‐censored failure data. The method is shown to be particularly useful when the sampling distribution is not known, which happens to be the case in a large number of practical reliability evaluations. Copyright © 2006 John Wiley & Sons, Ltd.     

Reliability analysis for electronic devices using beta‐Weibull distribution

Today, in reliability analysis, the most used distribution to describe the behavior of electronic products under voltage profiles is the Weibull distribution. Nevertheless, the Weibull distribution does not provide a good fit to lifetime datasets that exhibit bathtub‐shaped or upside‐down bathtub–shaped (unimodal) failure rates, which are often encountered in the reliability analysis of electronic devices. In this paper, a reliability model based on the beta‐Weibull distribution and the inverse power law is proposed. This new model provides a better approach to model the performance and fit of the lifetimes of electronic devices. To estimate the parameters of the proposed model, a Bayesian analysis is used. A case study based on the lifetime of a surface mounted electrolytic capacitor is presented, the results showed that the estimation of the proposed model differs from the inverse power law–Weibull and that it affects directly the mean time to failure, the failure rate, the behavior, and the performance of the capacitor under analysis.     

Reliability analysis of a ‘point‐and‐point machine’ of the Indian railway signaling system

The operations of one of the vital subsystems of Indian Railways (IR) signaling called a ‘point‐and‐point machine’, which guides a train in changing its direction of movement from one track to another, are affected by a number of problems in relation to its intensity of use, repair and maintenance, and environmental stress. In this paper, the reliability modeling of a point‐and‐point machine is elaborately described taking into account the effect of the problems on its operational efficiency. It is observed that even when derating the system specifications (i.e. lowering the speed of the train), the reliability of the point‐and‐point machine system has been poor (0.44 after 1100 days of continuous operation) with existing preventive and corrective measures. Two graphical methods, namely, Nelson–Aalen and total‐time‐on‐test plots, are used to analyze the trend of the failure time data for a set of representative point‐and‐point machines in the Kharagpur Division of IR. The parametric model as recommended for estimating the reliability of the point‐and‐point machine is capable of correctly predicting its failure rate probability. The analysis of data obtained from a homogeneous Poisson process of the point‐and‐point machine system also leads to a better assessment of the reliability pattern with which the problems as mentioned above may be adequately and properly addressed. Copyright © 2007 John Wiley & Sons, Ltd.     

Inference for a geometric‐poisson‐Rayleigh distribution under progressive‐stress accelerated life tests based on type‐I progressive hybrid censoring with binomial removals

Based on failures of a parallel‐series system, a new distribution called geometric‐Poisson‐Rayleigh distribution is proposed. Some properties of the distribution are discussed. A real data set is used to compare the new distribution with other 6 distributions. The progressive‐stress accelerated life tests are considered when the lifetime of an item under use condition is assumed to follow the geometric‐Poisson‐Rayleigh distribution. It is assumed that the scale parameter of the geometric‐Poisson‐Rayleigh distribution satisfies the inverse power law such that the stress is a nonlinear increasing function of time and the cumulative exposure model for the effect of changing stress holds. Based on type‐I progressive hybrid censoring with binomial removals, the maximum likelihood and Bayes (using linear‐exponential and general entropy loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the maximum likelihood, conditional median, best unbiased, and Bayes point predictors for future order statistics are obtained. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. Finally, a simulation study is performed, and numerical computations are performed to compare the performance of the implemented methods of estimation and prediction.     

Irregular Time‐Varying Stress Degradation Path Modeling: a Case Study on Lithium‐ion Cell Degradation

The aim of this paper is to investigate the issue of degradation modeling and reliability assessment for products under irregular time‐varying stresses. Conventional degradation models have been extensively used in the relevant literature to characterize degradation processes under deterministic stresses. However, the time‐varying stress, which may affect degradation processes, widely exists in field conditions. This paper extends the general degradation‐path model by considering the effects of time‐varying stresses. The new degradation‐path model captures influences of varying stresses on performance characteristics. A nonlinear least square method is used to estimate the unknown parameters of the proposed model. A bootstrap algorithm is adopted for computing the confidence intervals of the mean time to failure and percentiles of the failure‐time distribution. Finally, a case study of lithium‐ion cells is presented to validate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Classical and Bayesian inference on progressive‐stress accelerated life testing for the extension of the exponential distribution under progressive type‐II censoring

In this paper, a progressive‐stress accelerated life test under progressive type‐II censoring is considered. The cumulative exposure model is assumed when the lifetime of test units follows an extension of the exponential distribution. The maximum likelihood and Bayes estimates of the model parameters are obtained. The approximate and credible confidence intervals of the estimators are derived. Furthermore, a real lifetime data set is analyzed to illustrate the proposed procedures. Finally, the simulation studies are used to compare between 2 different designs of the progressive‐stress test (simple and multiple ramp‐stress tests).     

Comparison of Methods to Estimate the Time‐to‐failure Distribution in Degradation Tests

Degradation tests are alternative approaches to lifetime tests and accelerated lifetime tests in reliability studies. Based on a degradation process of a product quality characteristic over time, degradation tests provide enough information to estimate the time‐to‐failure distribution. Some estimation methods, such as analytical, the numerical or the approximated, can be used to obtain the time‐to‐failure distribution. They are chosen according to the complexity of the degradation model used in the data analysis. An example of the application and analysis of degradation tests is presented in this paper to characterize the durability of a product and compare the various estimation methods of the time‐to‐failure distribution. The example refers to a degradation process related to an automobile's tyre, and was carried out to estimate its average distance covered and some percentiles of interest. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

A competing risk model for the reliability of cylinder liners in marine Diesel engines

In this paper, a competing risk model is proposed to describe the reliability of the cylinder liners of a marine Diesel engine. Cylinder liners presents two dominant failure modes: wear degradation and thermal cracking. The wear process is described through a stochastic process, whereas the failure time due to the thermal cracking is described by the Weibull distribution. The use of the proposed model allows performing goodness-of-fit test and parameters estimation on the basis of both wear and failure data. Moreover, it enables reliability estimates of the state of the liners to be obtained and the hierarchy of the failure mechanisms to be determined for any given age and wear level of the liner. The model has been applied to a real data set: 33 cylinder liners of Sulzer RTA 58 engines, which equip twin ships of the Grimaldi Group. Estimates of the liner reliability and of other quantities of interest under the competing risk model are obtained, as well as the conditional failure probability and mean residual lifetime, given the survival age and the accumulated wear. Furthermore, the model has been used to estimate the probability that a liner fails due to one of the failure modes when both of these modes act.     

Fitting fatigue data with a bi‐conditional model

The formulation of a probability‐stress‐life (P‐S‐N) curve is a necessary step beyond the basic S‐N relation when dealing with reliability. This paper presents a model, relevant to materials that exhibits a fatigue limit, which considers the number of cycles to failure and the occurrence of the failure itself as statistically independent events, described with different distributions and/or different degree of scatter. Combining these two as a parallel system leads to the proposed model. In the case where the S‐N relation is a Basquin's law, the formulations of the probability density function, cumulative distribution function, quantiles, parameter and quantile confidence interval are presented in a procedure that allows practically any testing strategy. The result is a flexible model combined with the tools that deliver a wide range of information needed in the design phase. Finally, an extension to include static strength and applicability to fatigue crack growth and defects‐based fatigue approach are presented.