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.     

Integration method for reliability assessment with multi-source incomplete accelerated degradation testing data

ABSTRACT

During the product life cycle, the lifetime information will be collected at each stage, mainly from different tests at the R&D phase, field usage, and maintenance. To comprehensively conduct reliability assessments, it generally requires the integration of multi-source datasets, even that from similar products. In this article, we considered the scenario that products have been arranged with several accelerated degradation tests (ADT) under different types of accelerated stresses with dependency. The obtained data is called incomplete ADT dataset with incomplete stress conditions which fails the traditional integration method for reliability assessments. A novel method is proposed to accomplish this task through mutually exclusive set (MES) theory. The probability assignments for each dataset are given through the union set of several MESs. Then, the multi-source ADT datasets are integrated with the assigned weights of probabilities. Finally, a simulation study and a real application are given to illustrate the effectiveness of the proposed methodology.     

A sequential constant‐stress accelerated life testing scheme and its Bayesian inference

In the analysis of accelerated life testing (ALT) data, some stress‐life model is typically used to relate results obtained at stressed conditions to those at use condition. For example, the Arrhenius model has been widely used for accelerated testing involving high temperature. Motivated by the fact that some prior knowledge of particular model parameters is usually available, this paper proposes a sequential constant‐stress ALT scheme and its Bayesian inference. Under this scheme, test at the highest stress is firstly conducted to quickly generate failures. Then, using the proposed Bayesian inference method, information obtained at the highest stress is used to construct prior distributions for data analysis at lower stress levels. In this paper, two frameworks of the Bayesian inference method are presented, namely, the all‐at‐one prior distribution construction and the full sequential prior distribution construction. Assuming Weibull failure times, we (1) derive the closed‐form expression for estimating the smallest extreme value location parameter at each stress level, (2) compare the performance of the proposed Bayesian inference with that of MLE by simulations, and (3) assess the risk of including empirical engineering knowledge into ALT data analysis under the proposed framework. Step‐by‐step illustrations of both frameworks are presented using a real‐life ALT data set. Copyright © 2008 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.     

An approach that can quickly assess product reliability

One of the ways to determine the inherent reliability of a design is to test it under controlled environments based on the product usage that is understood by the development requirements. This can be accomplished by performing a reliability growth test on the product. A testing approach can be developed that enhances the product reliability and reduces the production testing cycle. Research performed to date points to the fact that this proposed methodology may not exist, and is the focus of continued research to refine the development of an approach to fill this gap. The combining of multiple testing approaches in order to ensure that the reliability requirement is met or exceeded while at the same time having the capability to reduce the testing cycle time when required due to schedule and cost constraints has not been addressed in the open literature till date. The methodology is to utilize a combination of multiple testing approaches to accomplish this task by exploring complementary testing ideas from various technologies that have been utilized previously with documented success. This approach demonstrated that component‐level testing reduced the product‐level failures by greater than 80% while at the same time reducing the schedule to complete all testing. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian multiple testing procedures for hotspot identification

Ranking a group of candidate sites and selecting from it the high-risk locations or hotspots for detailed engineering study and countermeasure evaluation is the first step in a transport safety improvement program. Past studies have however mainly focused on the task of applying appropriate methods for ranking locations, with few focusing on the issue of how to define selection methods or threshold rules for hotspot identification. The primary goal of this paper is to introduce a multiple testing-based approach to the problem of selecting hotspots. Following the recent developments in the literature, two testing procedures are studied under a Bayesian framework: Bayesian test with weights (BTW) and a Bayesian test controlling for the posterior false discovery rate (FDR) or false negative rate (FNR). The hypotheses tests are implemented on the basis of two random effect or Bayesian models, namely, the hierarchical Poisson/Gamma or Negative Binomial model and the hierarchical Poisson/Lognormal model. A dataset of highway–railway grade crossings is used as an application example to illustrate the proposed procedures incorporating both the posterior distribution of accident frequency and the posterior distribution of ranks. Results on the effects of various decision parameters used in hotspot identification procedures are discussed.     

A Bayesian generalized Eyring-Weibull accelerated life testing model

In this paper, a novel approach to a Bayesian accelerated life testing model is presented. The Weibull distribution is used as the life distribution and the generalized Eyring model as the time transformation function. This is a model that allows for the use of more than one stressor, whereas other commonly used acceleration models, such as the Arrhenius and power law models, incorporate one stressor. The use of the generalized Eyring-Weibull model developed in this paper is demonstrated in a case study, where Markov chain Monte Carlo methods are utilized to generate samples for posterior inference.     

A literature review on planning and analysis of accelerated testing for reliability assessment

Accelerated testing has been widely used for several decades. Started with accelerated life tests with constant‐stress loadings, more interest has been focused prominently on accelerated degradation tests and time‐varying stress loadings. Because accelerated testing is crucial to the assessment of product reliability and the design of warranty policy, it is important to develop an efficacious test plan that encompasses and addresses important issues, such as design of stress profiles, sample allocation, test duration, measurement frequency, and budget constraint. In recent years, extensive research has been conducted on the optimal design of accelerated testing plans, and the consideration of multiple stresses with interactions has become a big challenge in such experimental designs. The purpose of this study is to provide a comprehensive review of important methods for statistical inference and optimal design of accelerated testing plans by compiling the existing body of knowledge in the area of accelerated testing. In this work, different types of test planning strategies are categorized, and their drawbacks and the research trends are provided to assist researchers and practitioners in conducting new research in this area.     

A Bayesian analysis of beta testing

In this article, we define a model for fault detection during the beta testing phase of a software design project. Given sampled data, we illustrate how to estimate the failure rate and the number of faults in the software using Bayesian statistical methods with various different prior distributions. Secondly, given a suitable cost function, we also show how to optimize the duration of a further test period for each one of the prior distribution structures considered. Michael Wiper acknowledges assistance from the Spanish Ministry of Science and Technology via the project BEC2000-0167 and support from projects SEJ2004-03303 and 06/HSE/0181/2004     

Bayesian networks for multilevel system reliability

Bayesian networks have recently found many applications in systems reliability; however, the focus has been on binary outcomes. In this paper we extend their use to multilevel discrete data and discuss how to make joint inference about all of the nodes in the network. These methods are applicable when system structures are too complex to be represented by fault trees. The methods are illustrated through four examples that are structured to clarify the scope of the problem.     

A Bayes approach to reliability prediction utilizing data from accelerated life tests and field failure observations

A Bayes approach is proposed to improve product reliability prediction by integrating failure information from both the field performance data and the accelerated life testing data. It is found that a product's field failure characteristic may not be directly extrapolated from the accelerated life testing results because of the variation of field use condition that cannot be replicated in the lab‐test environment. A calibration factor is introduced to model the effect of uncertainty of field stress on product lifetime. It is useful when the field performance of a new product needs to be inferred from its accelerated life test results and this product will be used in the same environment where the field failure data of older products are available. The proposed Bayes approach provides a proper mechanism of fusing information from various sources. The statistical inference procedure is carried out through the Markov chain Monte Carlo method. An example of an electronic device is provided to illustrate the use of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Bayesian methodology for reliability model acceptance

This paper develops a methodology to assess the reliability computation model validity using the concept of Bayesian hypothesis testing, by comparing the model prediction and experimental observation, when there is only one computational model available to evaluate system behavior. Time-independent and time-dependent problems are investigated, with consideration of both cases: with and without statistical uncertainty in the model. The case of time-independent failure probability prediction with no statistical uncertainty is a straightforward application of Bayesian hypothesis testing. However, for the life prediction (time-dependent reliability) problem, a new methodology is developed in this paper to make the same Bayesian hypothesis testing concept applicable. With the existence of statistical uncertainty in the model, in addition to the application of a predictor estimator of the Bayes factor, the uncertainty in the Bayes factor is explicitly quantified through treating it as a random variable and calculating the probability that it exceeds a specified value. The developed method provides a rational criterion to decision-makers for the acceptance or rejection of the computational model.     

A model for Bayesian software reliability analysis

Two problems which are of great interest in relation to software reliability are the prediction of future times to failure and the calculation of the optimal release time. An important assumption in software reliability analysis is that the reliability grows whenever bugs are found and removed. In this paper we present a model for software reliability analysis using the Bayesian statistical approach in order to incorporate in the analysis prior assumptions such as the (decreasing) ordering in the assumed constant failure rates of prescribed intervals. We use as prior model the product of gamma functions for each pair of subsequent interval constant failure rates, considering as the location parameter of the first interval the failure rate of the following interval. In this way we include the failure rate ordering information. Using this approach sequentially, we predict the time to failure for the next failure using the previous information obtained. Using also the relevant predictive distributions obtained, we calculate the optimal release time for two different requirements of interest: (a) the probability of an in‐service failure in a prescribed time t; (b) the cost associated with a single or more failures in a prescribed time t. Finally a numerical example is presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Discrimination between accelerated life models via Approximate Bayesian Computation

Accelerated life testing (ALT) is widely used in high-reliability product estimation to get relevant information about an item's performance and its failure mechanisms. To analyse the observed ALT data, reliability practitioners need to select a suitable accelerated life model based on the nature of the stress and the physics involved. A statistical model consists of (i) a lifetime distribution that represents the scatter in product life and (ii) a relationship between life and stress. In practice, several accelerated life models could be used for the same failure mode and the choice of the best model is far from trivial. For this reason, an efficient selection procedure to discriminate between a set of competing accelerated life models is of great importance for practitioners. In this paper, accelerated life model selection is approached by using the Approximate Bayesian Computation (ABC) method and a likelihood-based approach for comparison purposes. To demonstrate the efficiency of the ABC method in calibrating and selecting accelerated life model, an extensive Monte Carlo simulation study is carried out using different distances to measure the discrepancy between the empirical and simulated times of failure data. Then, the ABC algorithm is applied to real accelerated fatigue life data in order to select the most likely model among five plausible models. It has been demonstrated that the ABC method outperforms the likelihood-based approach in terms of reliability predictions mainly at lower percentiles particularly useful in reliability engineering and risk assessment applications. Moreover, it has shown that ABC could mitigate the effects of model misspecification through an appropriate choice of the distance function.     

Assessing high reliability via Bayesian approach and accelerated tests

Sometimes the assessment of very high reliability levels is difficult for the following main reasons:

the high reliability level of each item makes it impossible to obtain, in a reasonably short time, a sufficient number of failures;

the high cost of the high reliability items to submit to life tests makes it unfeasible to collect enough data for ‘classical’ statistical analyses.

In the above context, this paper presents a Bayesian solution to the problem of estimation of the parameters of the Weibull–inverse power law model, on the basis of a limited number (say six) of life tests, carried out at different stress levels, all higher than the normal one.The over-stressed (i.e. accelerated) tests allow the use of experimental data obtained in a reasonably short time. The Bayesian approach enables one to reduce the required number of failures adding to the failure information the available a priori engineers' knowledge. This engineers' involvement conforms to the most advanced management policy that aims at involving everyone's commitment in order to obtain total quality.A Monte Carlo study of the non-asymptotic properties of the proposed estimators and a comparison with the properties of maximum likelihood estimators closes the work.     

Bayesian modeling and test planning for multiphase reliability assessment

We propose a Bayesian hierarchical model to assess the reliability of a family of vehicles, based on the development of the joint light tactical vehicle (JLTV). The proposed model effectively combines information across three phases of testing and across common vehicle components. The analysis yields estimates of failure rates for specific failure modes and vehicles as well as an overall estimate of the failure rate for the family of vehicles. We are also able to obtain estimates of how well vehicle modifications between test phases improve failure rates. In addition to using all data to improve on current assessments of reliability and reliability growth, we illustrate how to leverage the information learned from the three phases to determine appropriate specifications for subsequent testing that will demonstrate if the reliability meets a given reliability threshold.     

A novel Bayesian approach to reliability modeling: The benefits of uncertainty evaluation in the model selection procedure

This paper proposes a different likelihood formulation within the Bayesian paradigm for parameter estimation of reliability models. Moreover, the assessment of the uncertainties associated with parameters, the goodness of fit, and the model prediction of reliability are included in a systematic framework for better aiding the model selection procedure. Two case studies are appraised to highlight the contributions of the proposed method and demonstrate the differences between the proposed Bayesian formulation and an existing Bayesian formulation.     

A framework for improving the efficiency of operational testing through Bayesian adaptive design

When developing a product, it is important to consider product performance from a user perspective. This type of evaluation can be done through operational testing—assessing the ability of representative users to satisfactorily accomplish tasks with the product in operationally representative environments. This process can be expensive and time-consuming, but is critical to understanding whether the product can adequately do the job for which it was designed. We show how an existing design of experiments (DOEs) process for operational testing can be leveraged to construct a Bayesian adaptive design. This design, nested within the larger design created by the DOE process, allows interim analyses to stop testing early for success or futility. Representative simulations are presented to demonstrate how these interim analyses can be used in an operational test setting, and reductions in necessary test events are shown. The application of Bayesian-adaptive design methods will allow future operational testing to be conducted in less time and at less expense, on average, without compromising the ability of the existing process to verify the product meets the user's needs.     

A hierarchical modeling approach to accelerated degradation testing data analysis: A case study

In this case study, we investigate the degradation process of light‐emitting diodes (LEDs), which is used as a light source in DNA sequencing machines. Accelerated degradation tests are applied by varying temperature and forward current, and the light outputs are measured by a computerized measuring system. A degradation path model, which connects to the LED function recommended in Mitsuo (1991), is used in describing the degradation process. We consider variations in both measurement errors and degradation paths among individual test units. It is demonstrated that the hierarchical modeling approach is flexible and powerful in modeling a complex degradation process with nonlinear function and random coefficient. After fitting the model by maximum likelihood estimation, the failure time distribution can be obtained by simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian risk-based decision method for model validation under uncertainty

This paper develops a decision-making methodology for computational model validation, considering the risk of using the current model, data support for the current model, and cost of acquiring new information to improve the model. A Bayesian decision theory-based method is developed for this purpose, using a likelihood ratio as the validation metric for model assessment. An expected risk or cost function is defined as a function of the decision costs, and the likelihood and prior of each hypothesis. The risk is minimized through correctly assigning experimental data to two decision regions based on the comparison of the likelihood ratio with a decision threshold. A Bayesian validation metric is derived based on the risk minimization criterion. Two types of validation tests are considered: pass/fail tests and system response value measurement tests. The methodology is illustrated for the validation of reliability prediction models in a tension bar and an engine blade subjected to high cycle fatigue. The proposed method can effectively integrate optimal experimental design into model validation to simultaneously reduce the cost and improve the accuracy of reliability model assessment.