Optimal Design for Step-Stress Accelerated Degradation Test with Multiple Performance Characteristics Based on Gamma Processes

Step-stress accelerated degradation test (SSADT) plays an important role in assessing the lifetime distribution of highly reliable products under normal operating conditions when there are not enough test units available for testing purposes. Recently, the optimal SSADT plans are presented based on an underlying assumption that there is only one performance characteristic. However, many highly reliable products usually have complex structure, with their reliability being evaluated by two or more performance characteristics. At the same time, the degradation of these performance characteristics would be always positive and strictly increasing. In such a case, the gamma process is usually considered as a degradation process due to its independent and nonnegative increments properties. Therefore, it is of great interest to design an efficient SSADT plan for the products with multiple performance characteristics based on gamma processes. In this work, we first introduce reliability model of the degradation products with two performance characteristics based on gamma processes, and then present the corresponding SSADT model. Next, under the constraint of total experimental cost, the optimal settings such as sample size, measurement times, and measurement frequency are obtained by minimizing the asymptotic variance of the estimated 100 qth percentile of the product’s lifetime distribution. Finally, a numerical example is given to illustrate the proposed procedure.     

Optimal design of accelerated degradation tests based on Wiener process models

Optimal accelerated degradation test (ADT) plans are developed assuming that the constant-stress loading method is employed and the degradation characteristic follows a Wiener process. Unlike the previous works on planning ADTs based on stochastic process models, this article determines the test stress levels and the proportion of test units allocated to each stress level such that the asymptotic variance of the maximum-likelihood estimator of the qth quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are also developed for checking the validity of the relationship between the model parameters and the stress variable. Finally, using an example, sensitivity analysis procedures are presented for evaluating the robustness of optimal and compromise plans against the uncertainty in the pre-estimated parameter value, and the importance of optimally determining test stress levels and the proportion of units allocated to each stress level are illustrated.     

Optimal design for constant-stress accelerated degradation test based on gamma process

This paper addresses the optimal design problems for constant-stress accelerated degradation test (CSADT) based on gamma processes with fixed effect and random effect. For three optimization criteria, we prove that optimal CSADT plans with multiple stress levels degenerate to two-stress-level test plans only using the minimum and maximum stress levels under model assumptions. Under each optimization criterion, the optimal sample size allocation proportions for the minimum and maximum stress levels are determined theoretically. The effect of the stress level on the objective functions is also discussed. A numerical example and a simulation study are provided to illustrate the obtained results.     

Multiple-Steps Step-Stress Accelerated Degradation Modeling Based on Wiener and Gamma Processes

During the step-stress accelerated degradation test (SSADT) experiment, the operator usually elevates the stress level at a predetermined time-point for all tested products that had not failed. This time-point is determined by the experience of the operator and the test is carried on until the performance degradation value of the product crosses the threshold value. In fact, this mode only works when a lot of products have been used in the experiment. But in the SSADT experiment, the number of products is relatively few, and so the test control to the products should be done more carefully. Considering the differences in products, we think the time-point of elevating stress level varies randomly from product-to-product. We consider a situation in which when the degradation value crosses a pre-specified value, the stress level is elevated. Under the circumstances, the time at which the degradation path crosses the pre-specified value is uncertain, and so we may regard it as a random variable. We discuss multiple-steps step-stress accelerated degradation models based on Wiener and gamma processes, respectively, and we apply the Bayesian Markov chain Monte Carlo (MCMC) method for such analytically intractable models to obtain the maximum likelihood estimates (MLEs) efficiently and present some computational results obtained from our implementation.     

Joint modeling of linear degradation and failure time data with masked causes of failure under simple step-stress test

In this paper, we propose a method to model the relationship between degradation and failure time for a simple step-stress test where the underlying degradation path is linear and different causes of failure are possible. It is assumed that the intensity function depends only on the degradation value. No assumptions are made about the distribution of the failure times. A simple step-stress test is used to induce failure experimentally and a tampered failure rate model is proposed to describe the effect of the changing stress on the intensities. We assume that some of the products that fail during the test have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the maximum likelihood estimates of the model parameters are obtained through the expectation–maximization algorithm by treating the causes of failure as missing values. The effect of incomplete information on the estimation of parameters is studied through a Monte-Carlo simulation. Finally, a real-world example is analysed to illustrate the application of the proposed methods.     

Real-time reliability evaluation of two-phase Wiener degradation process

Reliability modeling and evaluation for the two-phase Wiener degradation process are studied. For many devices, the degradation rates could possibly increase or decrease in a non smooth manner at some point in time due to the change of degradation mechanism. A two-phase Wiener degradation process with an unobserved change point is used to model the degradation process. And we assume that the change point varies randomly from device to device. Furthermore, we integrate historical data and up-to-date observation data to improve the degradation modeling and evaluation based on Bayesian method. The change point between the two phases was obtained based on the Akaike information criterion (AIC) and the criterion of the residual sum of squares. Finally, a real example of liquid coupling devices (LCDs) and a numeric example are discussed to demonstrate the effectiveness of the proposed method. The results show that the proposed method is effective and efficient.     

Bivariate Constant-Stress Accelerated Degradation Model and Inference

To assess the reliability of highly reliable products that have two or more performance characteristics (PCs) in an accurate manner, relations between the PCs should be taken duly into account. If they are not independent, it would then become important to describe the dependence of the PCs. For many products, the constant-stress degradation test cannot provide sufficient data for reliability evaluation and for this reason, accelerated degradation test is usually performed. In this article, we assume that a product has two PCs and that the PCs are governed by a Wiener process with a time scale transformation, and the relationship between the PCs is described by the Frank copula function. The copula parameter is dependent on stress and assumed to be a function of stress level that can be described by a logistic function. Based on these assumptions, a bivariate constant-stress accelerated degradation model is proposed here. The direct likelihood estimation of parameters of such a model becomes analytically intractable, and so the Bayesian Markov chain Monte Carlo (MCMC) method is developed here for this model for obtaining the maximum likelihood estimates (MLEs) efficiently. For an illustration of the proposed model and the method of inference, a simulated example is presented along with the associated computational results.     

Failure Inference and Optimization for Step Stress Model Based on Bivariate Wiener Model

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distribution.     

Inverse Gaussian process models for bivariate degradation analysis: A Bayesian perspective

This article conducts a Bayesian analysis for bivariate degradation models based on the inverse Gaussian (IG) process. Assume that a product has two quality characteristics (QCs) and each of the QCs is governed by an IG process. The dependence of the QCs is described by a copula function. A bivariate simple IG process model and three bivariate IG process models with random effects are investigated by using Bayesian method. In addition, a simulation example is given to illustrate the effectiveness of the proposed methods. Finally, an example about heavy machine tools is presented to validate the proposed models.     

Residual life estimation based on bivariate non-stationary gamma degradation process

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method.     

Bayesian estimation of reliability using time-to-failure distribution of parametric degradation models

This paper focusses on computing the Bayesian reliability of components whose performance characteristics (degradation – fatigue and cracks) are observed during a specified period of time. Depending upon the nature of degradation data collected, we fit a monotone increasing or decreasing function for the data. Since the components are supposed to have different lifetimes, the rate of degradation is assumed to be a random variable. At a critical level of degradation, the time to failure distribution is obtained. The exponential and power degradation models are studied and exponential density function is assumed for the random variable representing the rate of degradation. The maximum likelihood estimator and Bayesian estimator of the parameter of exponential density function, predictive distribution, hierarchical Bayes approach and robustness of the posterior mean are presented. The Gibbs sampling algorithm is used to obtain the Bayesian estimates of the parameter. Illustrations are provided for the train wheel degradation data.     

Designing a Degradation Test with a Two-Parameter Exponential Lifetime Distribution

Degradation testing (DT) is a useful approach to assessing the reliability of highly reliable products which are not likely to fail under the traditional life tests or accelerated life tests. There have been a great number of excellent studies investigating the estimation of the failure time distribution and the optimal design (e.g., the optimal setting of the inspection frequency, the number of measurement, and the termination time) for DTs. However, the lifetime distributions considered in the studies mentioned above are all those without failure-free life. Here, failure-free life is characterized by a threshold parameter below which no failure is possible. The main purpose of this article is to deal with the optimal design of a DT with a two-parameter exponential lifetime distribution. More specifically, with respect to a DT where a linearized degradation model is used to model the degradation process and the lifetime is assumed to follow a two-parameter exponential distribution, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal combination of the inspection frequency, the sample size, and the termination time are determined by minimizing the mean squared error of the estimated 100p-th percentile of the lifetime distribution of the product. An example is provided to illustrate the proposed method and the corresponding sensitivity analysis is also discussed.     

Bivariate degradation analysis of products based on Wiener processes and copulas

Modern highly reliable products usually have complex structure and many functions. This means that they may have two or more performance characteristics. All the performance characteristics can reflect the product's performance degradation over time, and they may be independent or dependent. If the performance characteristics are independent, they can be modelled separately. But if they are not independent, it is very important to find the joint distribution function of the performance characteristics for estimating the reliability of the product as accurately as possible. Here, we suppose that a product has two performance characteristics and the degradation paths of these two performance characteristics can be governed by a Wiener process with a time-scale transformation, and that the dependency of the performance characteristics can be described by a copula function. The parameters of the two performance characteristics and the copula function can be estimated jointly. The model in such a situation is very complicated and analytically intractable and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum-likelihood estimates of the parameters to be determined in an efficient manner. For an illustration of the proposed model, a numerical example about fatigue cracks is presented.     

Planning sequential constant-stress accelerated life tests with stepwise loaded auxiliary acceleration factor

This article presents a design approach for sequential constant-stress accelerated life tests (ALT) with an auxiliary acceleration factor (AAF). The use of an AAF, if it exists, is to further amplify the failure probability of highly reliability testing items at low stress levels while maintaining an acceptable degree of extrapolation for reliability inference. Based on a Bayesian design criterion, the optimal plan optimizes the sample allocation, stress combination, as well as the loading profile of the AAF. In particular, a step-stress loading profile based on an appropriate cumulative exposure (CE) model is chosen for the AAF such that the initial auxiliary stress will not be too harsh. A case study, providing the motivation and practical importance of our study, is presented to illustrate the proposed planning approach.     

Optimal sequential Bayesian analysis for degradation tests

Degradation tests are especially difficult to conduct for items with high reliability. Test costs, caused mainly by prolonged item duration and item destruction costs, establish the necessity of sequential degradation test designs. We propose a methodology that sequentially selects the optimal observation times to measure the degradation, using a convenient rule that maximizes the inference precision and minimizes test costs. In particular our objective is to estimate a quantile of the time to failure distribution, where the degradation process is modelled as a linear model using Bayesian inference. The proposed sequential analysis is based on an index that measures the expected discrepancy between the estimated quantile and its corresponding prediction, using Monte Carlo methods. The procedure was successfully implemented for simulated and real data.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

On constant stress accelerated life tests terminated by Type II censoring at one of the stress levels

This paper considers constant stress accelerated life tests terminated by a Type II censoring regime at one of the stress levels. We consider a model based on Weibull distributions with constant shape and a log-linear link between scale and the stress factor. We obtain expectations associated with the likelihood function, and use these to obtain asymptotically valid variances and correlations for maximum likelihood estimates of model parameters. We illustrate their calculation, and assess agreement with observed counterparts for finite samples in simulation experiments. We then use moments to compare the information obtained from variants of the design, and show that, with an appropriate allocation of items to stress levels, the design yields better estimates of model parameters and related quantities than a single stress experiment.     

Optimal bivariate step-stress accelerated life test for Type-I hybrid censored data

This paper presents a step-stress accelerated life test for two stress variables to obtain optimal hold times under a Type-I hybrid censoring scheme. An exponentially distributed life and a cumulative exposure model are assumed. The maximum-likelihood estimates are given, from which the asymptotic variance and the Fisher information matrix are obtained. The optimal test plan is determined for each combination of stress levels by minimizing the asymptotic variance of reliability estimate at a typical operating condition. Finally, simulation results are discussed to illustrate the proposed criteria. Simulation results show that the proposed optimum plan is robust, and the initial estimates have a small effect on optimal values.     

Comparing distributions of time to onset of disease in animal tumorigenicity experiments

The cause-of-death test of Peto et al.(1980)pools information from a Hoel-Walburg test on incidental tumors with information from a logrank test on fatal tumors in order to compare the tumor rate of a group of rodents exposed to a carcinogen against the tumor rate of a group of unexposed animals. The cause-of-death test, which can arise as a partial likelihood score test from a model that assumes proportional odds for tumor prevalence and proportional hazards for tumor mortality, is not, in general, a direct test for equality of tumor onset distributions for occult tumors that are observed in both fatal and incidental contexts. This paper develops a direct cause-of-death test for comparing distributions of time to onset of occultumors. The test is derived as a partial likelihood score test under an assumed proportional hazards model for tumor onset distributions. The size and power of the proposed test are compared in a Monte Carlo simulation study to the size and power of competitive procedures, including procedures that do not require cause-of-death information.