Optimum Test Plans Research Articles

Design and Evaluation of Accelerated Life Testing Plans with Dual Objectives

Accelerated life testing (ALT) is widely used in industry to predict product lifetime and lifetime distribution. Optimal test plans based on statistical optimality criteria, such as D-optimality or Uc-optimality, have recently been explored in Monroe et al. (2010, 2011). However, in these studies, the product use condition was set at a constant; in reality, use stress may vary due to a changing product use environment. In this paper, we consider an I-optimality that minimizes the variance of lifetime prediction over the entire region of possible use conditions. In addition, we investigate the tradeoffs between model parameter estimation and model-based prediction for ALT test plans. Dual-objective optimal test plans are provided to experimenters so that they can make a decision to balance the plan's estimation and prediction properties. We employ some graphical tools, including fraction of use space (FUS) plot, efficiency plot, and Pareto frontier plot, to evaluate different test plans and to compare our proposed method with existing ones in literature.

Statistical Inference for a Simple Step-Stress Model Based on Censored Data from the Kumaraswamy Weibull Distribution

The step-stress accelerated life tests allow increasing the stress levels on test units at fixed time during the experiment. In this paper, accelerated life tests are considered when lifetime of a product follows a Kumaraswamy Weibull distribution. The shape parameter is assumed to be a log linear function of the stress and a cumulative exposure model holds. Based on Type II and Type I censoring, the maximum likelihood estimates are obtained for the unknown parameters. The reliability and hazard rate functions are estimated at usual conditions of stress. In addition, confidence intervals of the estimators are constructed. Optimum test plans are obtained to minimize the generalized asymptotic variance of the maximum likelihood estimators. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.

Discussion of Multistage Sampling Optimum Test Plans on Rice Cadmium De-tection for Jiangsu Province

Discussion of Multistage Sampling Optimum Test Plans on Rice Cadmium De-tection for Jiangsu Province

Truncated Life Test Plans for Economic Reliability Based on Four-Parametric Burr Distribution

Burr distribution is considered as a probability model for the lifetime of products. Reliability test plans are those sampling plans in which items from a lot are put to test to make conclusions on the estimate of life, and hence acceptance or rejection of the submitted lot is done. A test plan designs the termination time of the experiment and the termination number for a given sample size and producer’s risk. Tables and graphs were provided for certain specific values of designs, and it is useful to verify the optimum reliability test plan realized by Burr distributions.

Optimal design of accelerated life testing plans under progressive censoring

This article investigates the design of accelerated life testing plans under progressive censoring when test units experience competing failure modes and are subjected to either single or multiple stress types. The optimal test plan results in failure data at accelerated conditions that can then be used to obtain accurate reliability prediction at normal conditions. The new test plan criterion is based on the minimization of the asymptotic variance of the mean time of first failure and meets practical constraints. Numerical examples based on parameters from a real test are presented to illustrate the application of the proposed method.

On designing step-stress partially accelerated life tests under failure-censoring scheme

In this article the maximum likelihood estimates of the model parameters under step-stress partially accelerated life tests (SSPALT) are obtained assuming the Weibull distribution with Type-II censored data. Also, the confidence bounds of the parameters are obtained. In addition, optimum step stress test plans are developed. The optimum test plan determines the optimal stress change point that minimizes the generalized asymptotic variance of the maximum likelihood estimators for the model parameters. That is, improving the quality of the statistical inference.

Design of accelerated life testing plans under multiple stresses

Accelerated life testing (ALT) using multiple stresses is commonly used in practice to resemble the operating stresses at normal operating conditions and obtain failure observations in a much shorter time. However, to date, there is little research into the theory of planning ALT for reliability estimation with multiple stresses. ALT with multiple stresses can result in a large number of stress-level combinations which presents a challenge for implementation. In this article, we propose an approach for the design of ALT plans with multiple stresses and formulate multistress test plans based on different objectives and practical constraints. We develop a simulated annealing algorithm to efficiently determine the testing plan parameters. We demonstrate the proposed method with examples based on an actual test conducted using three stress types. The obtained optimal test plans are compared with those based on fractional factorial design. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 468–478, 2013

Optimum multi-objective modified constant-stress accelerated life test plan for the Burr type-XII distribution with type-I censoring

Accelerated life test is usually performed using constant, step, progressive, cyclic and random stress loadings. In most of the existing accelerated life test plans, the stress level changes instantaneously, which is impractical. So, it is desirable to increase the stress at some finite rate, since a sudden jump in stress level may cause a thermal shock or undesirable failure modes that may not occur at the normal use condition. To overcome this problem, modified constant-stress loading and modified step-stress loading have been used in the literature in each of which stress is increased from a low stress level to higher stress level at a finite rate. This article deals with the design of an optimum multi-objective modified constant-stress accelerated life test plan for Burr type-XII distribution with type-I censoring. The inverse power law and a cumulative exposure model are assumed. The optimal test plan consists of determining an optimum low stress level, and also optimum proportion of units to be allocated at low and high stress levels by minimizing the weighted sum of the asymptotic variances of the maximum likelihood estimator of quantile lifetimes at design constant stress. The multi-objective modified constant-stress accelerated life test plan devised has been explained using a numerical example and the effects of the deviations from the pre-estimates of the design parameters investigated. The proposed model has been compared with the pre-existing model using the simulated failure time data set. For the same data set the proposed multi-objective plan has been compared with a single-objective plan with respect to changes in the values of the parameters, equal weights and unequal weights.

Optimal Design for Accelerated Destructive Degradation Tests

Degradation tests are powerful and useful tools for lifetime assessment of highly reliable products. In some applications, the degradation measurement process would destroy the physical characteristic of units when tested at higher than usual stress levels of an accelerating variable such as temperature, so that only one measurement can be made on each tested unit during the degradation testing. An accelerated degradation test giving rise to such a degradation data is called an accelerated destructive degradation test (ADDT). The specification of the size of the total sample, the frequency of destructive measurements, the number of measurements at each stress level, and other decision variables are very important to plan and conduct an ADDT efficiently. A wrong choice of these decision variables may not only result in increasing the experimental cost, but may also yield an imprecise estimate of the reliability of the product at the use condition. Motivated by a polymer data, this article deals with the problem of designing an ADDT with a nonlinear model. Under the constraint that the total experimental cost does not exceed a pre-fixed budget, the optimal test plan is obtained by minimizing the asymptotic variance of the estimated 100 p th percentile of the product’s lifetime distribution at the use condition. A sensitivity analysis is also carried out to examine the effects of changes in the decision variables on the precision of the estimator of the 100 p th percentile. A simulation study further shows that the simulated values are quite close to the asymptotic values when the sample sizes are large enough.

Optimum Step Stress Accelerated Life Testing for Mukherjeeislam Failure Model

Step stress accelerated life testing model is presented to obtain the optimal hold time at which the stress level is changed. A Mukherjee-Islam Failure model is assumed for the failure time at any constant stress level and a cumulative exposure model is assumed. The scale parameter of the Mukherjee-Islam Failure model at constant stress levels is assumed to be a log-linear function of the stress level. The maximum likelihood function is given for the step stress accelerated life testing. Optimal SSALT plan is proposed by minimizing the Asymptotic Variance (AV) of the desired life reliability estimate. Fisher information matrix is obtained. An optimal test plan with the minimum asymptotic variance of reliability estimate at time τ is determined.

Optimum Multi-Objective Ramp-Stress Accelerated Life Test with Stress Upper Bound for Burr Type-XII Distribution

This paper deals with the formulation of optimum multi-objective ramp-stress accelerated life test (ALT) plans with stress upper bounds for the Burr type-XII distribution under type-I censoring. It is impractical to estimate only one objective parameter after conducting such costly ALT tests. Therefore, an optimum ramp-stress ALT plan with multiple estimating objectives has been designed. The Burr type-XII life distribution has been found appropriate in accelerated life testing models. The commonly used Weibull and exponential distributions are its limiting cases, and the log-logistic distribution is its particular case. The inverse power law, and a cumulative exposure model are assumed. The optimal test plan consists of determining the ramp rate by minimizing the weighted sum of the asymptotic variances of the maximum likelihood estimator of quantile lifetimes at design constant stress. The method developed has been illustrated using an example, and the effects of the pre estimates of design parameters were investigated. Comparative study has also been done to highlight the merits of the proposed model.

Optimum Design for Type-I Step-stress Accelerated Life Tests of Two-parameter Weibull Distributions

In this article, we focus on the general k-step step-stress accelerated life tests with Type-I censoring for two-parameter Weibull distributions based on the tampered failure rate (TFR) model. We get the optimum design for the tests under the criterion of the minimization of the asymptotic variance of the maximum likelihood estimate of the pth percentile of the lifetime under the normal operating conditions. Optimum test plans for the simple step-stress accelerated life tests under Type-I censoring are developed for the Weibull distribution and the exponential distribution in particular. Finally, an example is provided to illustrate the proposed design and a sensitivity analysis is conducted to investigate the robustness of the design.

Optimal design of multiple stress constant accelerated life test plan on non-rectangle test region

For optimal design of constant stress accelerated life test(CSALT) with two-stress, if the stresses could not reach the highest levels simultaneously, the test region becomes non-rectangular. For optimal CSALT design on non-rectangle test region, the present method is only focused on non-rectangle test region with simple boundary, and the optimization algorithm is based on experience which can not ensure to obtain the optimal plan. In this paper, considering the linear-extreme value model and the optimization goal to minimize the variance of lifetime estimate under normal stress, the optimal design method of two-stress type-I censored CSALT plan on general non-rectangular test region is proposed. First, two properties of optimal test plans are proved and the relationship of all the optimal test plans is determined analytically. Then, on the basis of the two properties, the optimal problem is simplified and the optimal design method of two-stress CSALT plan on general non-rectangular test region is proposed. Finally, a numerical example is used to illustrate the feasibility and effectiveness of the method. The result shows that the proposed method could obtain the optimal test plan on non-rectangular test regions with arbitrary boundaries. This research provides the theory and method for two-stress optimal CSALT planning on non-rectangular test regions.

Optimum Step Stress Accelerated Life Testing for Power Function Distribution

The optimum times of varying stress level for simple stress plan under a cumulative exposure model using the Power Function distribution are determined for a wide range of values of the model parameters. The scale parameter of the power function distribution at constant level is assumed to be a log linear function of the stress levels. The optimum test plan is obtained by minimizing with respect to the change time, the asymptotic variance (AV) of the maximum likelihood estimator (MLE) of a given 100p—th percentile of the distribution at the design stress.

OPTIMUM MULTI-LEVEL RAMP-STRESS ALT PLAN WITH MULTIPLE-OBJECTIVES FOR BURR TYPE-XII DISTRIBUTION UNDER TYPE-I CENSORING

This paper considers optimal design for ramp-stress accelerated life test (ALT) with multiple stresses and multiple estimating objectives using Burr type-XII life distribution and Type-I censoring. It is impractical to estimate only one objective parameter after conducting such costly ALT tests, therefore, an optimum ramp-stress ALT plan with multiple estimating objectives has been formulated. The need to analyze an ALT data with different life distribution models such as exponential, normal, Weibull, log logistic, etc., is necessitated since the use of correct life distribution model especially in the presence of a limited source of data-as typically occurs with modern devices, having high reliability, helps in preventing the choice of unnecessary and expensive planned replacements. The Burr type-XII distribution has a nonmonotone hazard function, which can accommodate many shapes of hazard function. The commonly used Weibull and exponential distributions are its limiting cases and log logistic distribution is its particular case. The distribution has been found appropriate for modeling failures that occur with less frequency and also when there is high occurrence of early failures. The inverse power law and a cumulative exposure model are assumed. The optimal test plan chooses the stress rates and proportion of units allocated to each stress by minimizing the weighted sum of the asymptotic variances of the maximum likelihood estimator of quantile lifetimes at design constant stress. The method developed has been illustrated using an example, sensitivity analysis carried out and comparative study has also been done to highlight the merits of the proposed model.

Bayesian Methods for Accelerated Destructive Degradation Test Planning

Accelerated Destructive Degradation Tests (ADDTs) provide timely product reliability information in practical applications. This paper describes Bayesian methods for ADDT planning under a class of nonlinear degradation models with one accelerating variable. We use a Bayesian criterion based on the estimation precision of a specified failure-time distribution quantile at use conditions to find optimum test plans. A large-sample approximation for the posterior distribution provides a useful simplification to the planning criterion. The general equivalence theorem (GET) is used to verify the global optimality of the numerically optimized test plans. Optimum plans usually provide insight for constructing compromise plans which tend to be more robust, and practically useful. We present a numerical example with a log-location-scale distribution to illustrate the Bayesian test planning methods, and to investigate the effects of the prior distribution and sample size on test planning results.

Optimum Degradation Tests for Comparison of Products

Life tests are often conducted to compare products, designs, or processes to select the one that has the highest reliability. To make a sound decision, engineers may require the tests to continue until all or most of the test units fail. As such, the test time is often excessive, especially when we test high-reliability products. For some products, a failure is said to have occurred when a performance characteristic reaches a specified threshold. For these products, reliability is defined by the performance characteristic. Then comparing the reliabilities of two products can be turned into comparison of their performance characteristics. During testing, it is possible to measure the performance characteristics at different times. Measurement data can be employed to predict whether or not the performance characteristics of the two products will be significantly different at the time of interest, e.g., design life, warranty time, or specified test time. When there are sufficient data to make such a prediction with a high degree of confidence, the test can be terminated. As a result, the test time is reduced. This paper presents a method for comparing the performance characteristics of two products with a reduced test time, where the log performance characteristics are -normally distributed with a mean as a (transformed) linear function of time, and a constant standard deviation. In particular, the paper describes a test method, test termination rules, and degradation modeling. Then we develop an overlap probability as a metric to measure the difference between two products. The confidence interval for the overlap probability is calculated. The reliability and its confidence interval are computed from the degradation data. The paper also delineates optimum test plans which choose sample size and test time to minimize the variance of an estimate related to the overlap probability, and simultaneously satisfy the constraints on test budget and available sample size. The test plans are robust against the pre-estimates of model parameters. The proposed method is illustrated with a practical example. The application shows that the method is effective at reducing test time.

Optimum constant-stress life test plans for Pareto distribution under type-I censoring

The paper considers the case of constant-stress partially accelerated life testing (CSPALT) when two stress levels are involved under type-I censoring. The lifetimes of test items are assumed to follow a two-parameter Pareto lifetime distribution. Maximum-likelihood method is used to estimate the parameters of CSPALT model. Confidence intervals for the model parameters are constructed. Optimum CSPALT plans that determine the best choice of the proportion of test units allocated to each stress are developed. Such optimum test plans minimize the generalized asymptotic variance of the maximum-likelihood estimators of the model parameters. For illustration, Monte Carlo simulation studies are presented.

Optimal Reliability Demonstration Test Plans for $k$-out-of-$n$ Systems of Gamma Distributed Components

Constrained optimization problems are solved to determine the best reliability sampling plans for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex> </formula> -out-of- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> </formula> :F systems of gamma distributed components. As usual, the constraints are related to the quality levels required by the producer and the consumer, whereas the decision criterion is based on the uniformly most powerful reliability test. Excellent approximate solutions are derived in closed-forms assuming a frequentist perspective. This classical methodology is generalized to those situations in which the prior information on the fraction effective, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r$</tex></formula> , is appreciable. Because prior knowledge on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r$</tex> </formula> often suggests a restricted parameter interval, a limited beta distribution is used to describe the probabilistic behavior of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$r$</tex></formula> . Optimal sample sizes and acceptance factors are obtained by solving mixed integer nonlinear programming problems. Lower and upper bounds on the smallest sample size, as well as a quite accurate approximation, are also provided. An efficient step-by-step procedure, which only needs a few iterations in most practical cases, is proposed for solving the optimization problem. The incorporation of prior knowledge into the reliability analysis provides better assessment of the true producer's and consumer's risks, and considerable savings in sample size and test duration. The optimal solutions are usually quite robust to small variations in the gamma shape parameter and the prior information. An example concerning a system composed of five water pumps for cooling a reactor is included for illustration. The proposed component test sampling plans can be recommended for use in practical applications, particularly when it is economically infeasible, or even impossible, to test a whole system.

English

&nbsp; &nbsp; This study discusses the point and interval estimations of two-parameter Gompertz distribution under partially accelerated life tests with Type-II censoring. Also, the study considers the optimal design problem in the case of time step-stress model. The maximum likelihood estimators and approximate confidence intervals of the model parameters are derived. Also, optimum test plans for the time step-stress life tests are developed. For illustration, a simulation study is provided. Key words:&nbsp;Gompertz distribution, partially accelerated life test, maximum likelihood estimation, Newton-Raphson method, fisher information matrix, Monte Carlo simulation, type-II censoring, optimal design, generalized asymptotic variance. &nbsp;