Designing an accelerated degradation experiment with a reciprocal Weibull degradation rate

Degradation tests are widely used to assess the reliability of highly reliable products which are not likely to fail under traditional life tests or accelerated life tests. However, for some highly reliable products, the degradation may be very slow and hence it is impossible to have a precise assessment within a reasonable amount of testing time. In such cases, an alternative is to use higher stresses to extrapolate the product's reliability at the design stress. This is called an accelerated degradation test (ADT). In conducting an ADT, several decision variables, such s the inspection frequency, sample size and termination time, at each stress level are influential on the experimental efficiency. An inappropriate choice of these decision variables not only wastes experimental resources but also reduces the precision of the estimation of the product's reliability at the use condition. The main purpose of this paper is to deal with the problem of designing an ADT. By using the criterion of minimizing the mean‐squared error of the estimated 100$p$th percentile of the product's lifetime distribution at the use condition subject to the constraint that the total experimental cost does not exceed a predetermined budget, a nonlinear integer programming problem is built to derive the optimal combination of the sample size, inspection frequency and the termination time at each stress level. A numerical example is provided to illustrate the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.

Assessing the reliability of a product is very important to improve the product’s quality and to get the trust of customers. Degradation experiments are usually used to assess the reliability of highly reliable products,which are not expected to fail under the traditional life tests. Several decision variables, such as the sample size, the inspection frequency, and the termination time, have a direct influence on the experimental cost and the estimation precision of lifetime information. This paper deals with the optimal design of a degradation experiment where the degradation rate follows a log-logistic distribution. Under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal decision variables are obtained by minimizing the mean squared error of the estimated 100pth percentile of thelifetime distribution of the product. A simulation study and a real example of drug potency data are provided to illustrate the proposed method.

Degradation experiments are widely used to assess the reliability of highly reliable products which are not likely to fail under the traditional life tests. In order to conduct a degradation experiment efficiently, several factors, such as the inspection frequency, the sample size, and the termination time, need to be considered carefully. These factors not only affect the experimental cost, but also affect the precision of the estimate of a product's lifetime. In this paper, we deal with the optimal design of a degradation experiment. Under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal decision variables are solved by minimizing the variance of the estimated 100pth percentile of the lifetime distribution of the product. An example is provided to illustrate the proposed method. Finally, a simulation study is conducted to investigate the robustness of this proposed method. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 689–706, 1999

This paper discusses educational tools, resources and experience in teaching physics-of-failure (PoF) modeling and data analysis methods used in accelerated life testing (ALT) and accelerated degradation testing (ADT). The concepts introduced in this paper include several probabilistic and highly technical approaches such as the PoF and mechanistic approaches to reliability prediction and assessments as well as advanced Bayesian analysis methods used in ALT and ADT to analyze life data obtained through various methods of constant and variable stress testing efforts. Other resources include codes and algorithms from the readily available computational tools R, MATLAB, and WinBUGS for solving advanced model and life estimation methods related to PoF, ALT and ADT analyses. This paper provides an overview of these resources and several example problems.

Estimating the long term performance of highly reliable products has been a difficult problem as accelerated life testing (ALT), which involves testing at highly elevated stresses, often results in too few failures for drawing useful inferences. To overcome this problem, accelerated degradation testing (ADT) has been proposed as a means to predict performance for highly reliable products. It requires one to identify a performance measure that would exhibit degradation and to monitor it over time. Product reliability can then be inferred from the degradation paths without the need of observing actual failures. Although physical failures are not needed in ADT, one usually defines failure as the first time when the degradation process exceeds a pre-specified threshold, so that the degradation path can be correlated to product reliability. As a result, reliability information of a product is embedded in degradation paths of units tested under ADT. In this paper, we look into planning of an ADT in which the test stress is increased in steps from lower stress to higher stress during the test, so that specimens are gradually conditioned to the stressed environment thus avoiding over-stressing. Our objective is to minimize the cost of testing, which is a function of sample size, test duration and number of inspections, as well as obtaining a reliability estimate of a requisite level of precision. Data from degradation paths are used to characterize the appropriate stochastic model underlying the product degradation process. We then derive the maximum likelihood estimators and the mean life at the use stress and its asymptotic variance. This variance is then used as a constraint, in a test plan which minimizes the testing cost. The optimal test plan gives the optimal sample size, number of inspections at each intermediate stress level and number of total inspections.

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.

With the increasing marketplace demand for decreasing test time and cost, constant‐stress accelerated life testing (CSALT) is widely used for reliability demonstration in practice. The existing accelerated acceptance sampling plans (AASPs) do not consider the uncertainty of accelerated factor (AF), which makes it difficult to accurately describe the producer's and consumer's risks. Previous AASP studies on CSALT were mainly carried out under a fixed accelerated stress level, and the influence of accelerated stress level is not concerned. Therefore, considering the uncertainty of AF and taking the lognormal distribution as an example, the AASP design method with time‐censoring is proposed, where the life‐stress relationship between the location parameter and stress level is expressed by an inverse power law model. First, considering the uncertainty of AF expressed by a distribution, the acceptance probability can be obtained by the mathematic expectation of conditional acceptance probability, and the producer's and consumer's risks can be accurately expressed. Then, to reduce the total cost furtherly, the accelerated stress level is considered as a new decision variable, the optimum AASP including the sampling plan and the decision‐making criterion can be designed by minimizing the total cost and solving the risk constraint equations of both sides. A real case study of white organic light‐emitting diodes is carried out to verify the effectiveness of the proposed method.

For the highly reliable products, an accelerated degradation test (ADT) is a useful tool which has been employed in industry to obtain reliability-related information within an affordable amount of time and cost. In an ADT, as all other reliability tests, it is important to carefully design the ADT beforehand to obtain estimates of the quantities of interest as precisely as possible. In this paper, optimal ADTs are developed assuming that the constant-stress loading method is employed and the degradation characteristic follows a Wiener process. Under the constraint that the total cost does not exceed a pre-specified budget, the stress levels, the number of test units allocated to each stress level and the number of measurement (termination time) are determined such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized.

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When standardized, the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation.

As the rapid development of modern technology, industrial companies have to manufacture high-reliability and long-lifetime products to satisfy the changing needs of the customers. In such cases, it is difficult to assess the reliability and lifetime of the product with traditional life testing. Thus how to evaluate the lifetime and reliability index of these products is an urgent problem to be solved. For some products, it is possible to obtain the performance degradation information by the accelerated degradation testing (ADT). Therefore, to utilize the product degradation information may be an effective way to resolve this issue. However, most of lifetime prediction methods in use are mainly based on single prediction model with the disadvantage of low robustness and inaccuracy, so it cannot get the high credible lifetime of the product. This paper will adopt the prediction thought of combining two single models to establish a new prediction model, also propose a combined life prediction method based on the performance degradation data by using the particle swarm optimization (PSO) with immunity algorithms (IA). The author selects two better prediction models, which are time series model and the model of Brownian motion with drift, to predict the degradation path of the product separately. As the degradation process of the product is stochastic, and these two models are also well suited for this process. So each of them is good for long-term prediction. Then the objective function of the sum of the absolute errors is established with weight coefficients which are used to combine these prediction methods and integrate the prediction results. And the objective function is minimized with the IA-PSO, which has the characteristics of fast global optimization and uses to determine the weight coefficients for each of the prediction models. This approach can reduce the blindness of calculation and improve the accuracy of the prediction model. To verify this combined prediction method, the application of SLD's performance degradation data is demonstrated in this paper, the result shows that the combined prediction method based on the IA-PSO is effective and reasonable, and evidently has less prediction errors than each of single model. Therefore, this method plays an instructive way in engineering practice and research field.

Purpose Zero-failure reliability testing aims at demonstrating whether the product has achieved the desired reliability target with zero failure and high confidence level at a given time. Incorporating accelerated degradation testing in zero-failure reliability demonstration test (RDT) facilitates early failure in high reliability items developed within short period of time to be able to survive in fiercely competitive market. The paper aims to discuss these issues. Design/methodology/approach The triangular cyclic stress uses one test chamber thus saving experimental cost. The parameters in model are estimated using maximum likelihood methods. The optimum plan consists in finding out optimum number of cycles, optimum specimens, optimum stress change point(s) and optimum stress rates. Findings The optimum plan consists in finding out optimum number of cycles, optimum specimens, optimum stress change point(s) and optimum stress rates by minimizing asymptotic variance of estimate of quantile of the lifetime distribution at use condition subject to the constraint that total testing or experimental cost does not exceed a pre-specified budget. Confidence intervals of the design parameters have been obtained and sensitivity analysis carried out. The results of sensitivity analysis show that the plan is robust to small deviations from the true values of baseline parameters. Originality/value For some highly reliable products, even accelerated life testing yields little failure data of units in a feasible amount of time. In such cases accelerated degradation testing is carried out, wherein the failure termed as soft failure is defined in terms of performance characteristic of the product exceeding its critical (threshold) value.

Quantifying inequality in the leaf area distribution within a single module is critical for elucidating plant resource allocation strategies, but the accuracy of theoretical Gini coefficients derived from statistical distributions remains poorly validated against observed values. To resolve this gap, we analyzed 9,242 leaves from 121 culms of the bamboo Semiarundinaria densiflora, a model system with minimal ontogenetic noise and moderate leaf counts (36–187 leaves per culm) that enables robust Lorenz curve construction. Four candidate distributions were tested: the normal, log-normal, two-parameter Gamma, and two-parameter Weibull distributions. The parameters of the normal and log-normal distributions were estimated directly from sample statistics, whereas the parameters of the Gamma and Weibull distributions were estimated using the maximum likelihood method. Goodness of fit was assessed using the Kolmogorov-Smirnov (K-S) test for distributional validity, and the Akaike’s information criterion (AIC) for model selection. Although the Gamma distribution passed the K-S test for a slightly higher percentage of culms (99%) than the Weibull distribution (97.5%), the Weibull distribution was selected as the superior model because it yielded significantly lower AIC values. Crucially, the theoretical Gini coefficients of the Gamma and Weibull distributions (denoted as GG and GW, respectively) were tested against the observed Gini coefficients (GP) calculated nonparametrically using the polygon method. Linear regression demonstrated that GW​ predicted GP with near isometric accuracy: the intercept’s 95% confidence interval included zero (−0.006 to 0.017) and the slope’s 95% confidence interval included unity (0.929 to 1.039). In contrast, GG exhibited significant bias. Notably, pooling leaves across culms violated all distributions due to microhabitat driven multimodality, confirming that intra-culm inequality assessments require organism level analysis. This work provides an empirical validation that the Weibull shape parameter reliably quantifies intra-culm leaf area inequality. By bridging theoretical distribution models with field-derived inequality metrics, our approach provides insights into canopy efficiency, photosynthetic optimization, and hydraulic trade-offs. Future work should test this approach in other grass species and assess its generalizability in plants with contrasting canopy architectures.

The author presents a study on the development of approximate theoretical forms of the distributions of wave steepness and surf parameter. The approach is based on assuming the random wave process to be long crested and narrow banded. The results are compared with data from measurements at sea representing two severe storms. Both parameters are lognormal distributed. The resulting statistics for the surf parameter are applied to breakers at normal incidence on sloping beaches. As stated by the author, the joint statistics of wave steepness with wave heights or crest heights, or the wave steepness with wave heights or crest heights above a specified threshold may be more appropriate in formulating risks of capsizing of vessels, overtopping, and slamming forces on seawalls, etc. However, the wave steepness is of interest by itself, particularly in relation with many of the surf-zone processes. This discussion points out that other data sets may result in other distributions. This is exemplified by using the data referred to by the author—the data used by Myrhaug and Kjeldsen 1984, 1987 , Myrhaug and Rue 1993 , and Myrhaug and Kvalsvold 1995 . These papers discuss various aspects of wave steepness statistics using data from a large population of waves obtained by waverider buoys at three different deep water locations on the Norwegian continental shelf. Fig. 1 here shows the cdf of the normalized wave steepness s=S /Srms in Weibull scale, where Srms is as defined in Eq. 20 of the original paper. The pluses represent the Myrhaug and Kjeldsen 1984 data. The full line represents the combined lognormal and Weibull distribution obtained as a best fit by eye to the data; the pdf of the lognormal distribution is given by Eq. 17 of the original paper with , = −0.257,0.523 for s 1.2; the pdf of the Weibull distribution is given by Eq. 14 with , = 1.40,0.84 for s 1.2. It appears that this model gives a good representation of the data. Myrhaug and Rue 1993 used the Weibull model to study the statistics of two successive wave steepness parameters with the focus on steep waves S 0.10 . Myrhaug and Kjeldsen 1984 , Myrhaug and Kvalsvold 1995 , and Myrhaug 1994 discussed the joint distribution of wave height and wave steepness by taking f h ,s = f s h f h where f s h is given by the lognormal distribution in Myrhaug and Kjeldsen 1984, Eq. 4 , with x=s, Eq. 18 , and Eq. 19 , and f h is given by the Weibull distribution in Myrhaug and Kjeldsen

This article deals with the design of optimal ramp–soak–stress accelerated degradation test using Wiener process for modeling degradation paths. In an accelerated degradation test, failure occurs when the performance characteristic crosses the critical value the first time. Accelerated degradation test is incorporated in zero-failure reliability demonstration test to facilitate reduction in the test duration resulting in reduction in the test cost of high-reliability items. The parameters in model are estimated using method of maximum likelihood. The optimum plan consists in finding out optimum number of ramp–soak–stress cycles, optimum number of specimens, optimum stress change point(s), and optimum stress rates by minimizing asymptotic variance of estimate of [Formula: see text] quantile of the lifetime distribution at use condition subject to the constraint that total testing or experimental cost does not exceed a prespecified budget. An example is given to illustrate the proposed model, and sensitivity analysis is also carried out.

Accelerated Degradation Testing (ADT) is proposed as a means to obtain degradation data of products in a short time period and extrapolate the lifetime and reliability of products under use condition. There are two optimality criterion could be defined in designing an accelerated test plan. Criterion D : some testing planners are interest in the prediction of products' lifetime, so their objective is minimizing the variance (or asymptotic variance) of the MLE of parameter Θ which can include the quantiles and the mean of the lifetime distribution at a pre-specified stress level. Criterion II : if engineers are interest in the estimation of model parameters, their objective may be maximizing the determinant of the Fisher information matrix, which has a reciprocal relationship with variance-covariance matrix. This criterion is D-optimality, wherein the `information' is maximized and the `variance' is minimized at the same time. Criterion I for Step Stress Accelerated De gradation Testing (SSADT) planning is already proposed, however Criterion II (D-optimality) is seldom discussed. In this study, a method to optimal design for SSADT based on D-optimality is present. First, drift Brownian motion is applied to describe a typical SSADT problem. Next, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal plan gives sample size, total testing time, stress levels and testing time at each stress level. Finally, an example is used to illustrate this test planning method.