Optimum Test Plans Research Articles

Planning reliability demonstration tests with limited expected risks

Optimal reliability demonstration test plans with controlled expected producer and consumer risks are computed. Device failure times follow Weibull distributions, whereas prior information on device reliability is described by limited beta models. Minimum-cost test durations, sample sizes and acceptance numbers are found by solving mixed integer nonlinear programming problems. Lower and upper bounds on the minimal feasible number of failures allowed, as well as a reasonable approximation, are first deduced. An iterative method is then presented for determining the optimal lot inspection plan. Numerical and graphical analyses and several applications in industrial manufacturing are also provided for illustrative and comparative purposes. The inclusion of substantial prior knowledge allows the reliability engineer to reduce test time, sample size and cost, and leads to improved risk assessments for both producers and consumers. The proposed approach also enables the practitioners the flexibility to define precise limits of the device reliability and to combine multiple experts’ opinions. In addition, the best plan can continually be updated as more subjective or objective information becomes available.

Optimal Plan and Estimation for Bivariate Step-Stress Accelerated Life Test under Progressive Type-I Censoring

In this paper, a step-stress accelerated life test with two stress variables for Weibull distribution under progressive type-I censoring is considered. The stress-life relationship as a log-linear function of stress levels, and for each combination of stress levels, a cumulative exposure model is assumed. The maximum likelihood and Bayes estimates of the model parameters are obtained. The optimum test plan is developed using variance-optimality criterion, which consists in finding out the optimal stress change time by minimizing asymptotic variance of the maximum likelihood estimates of the log of the scale parameter at the design stress. The proposed study illustrated by using simulated data.

MCMC Method for Exponentiated Lomax Distribution based on Accelerated Life Testing with Type I Censoring

Accelerated Life Testing (ALT) is an effective technique which has been used in different fields to obtain more failures in a shorter period of time. It is more economical than traditional reliability testing. In this article, we propose Bayesian inference approach for planning optimal constant stress ALT with Type I censoring. The lifetime of a test unit follows an exponentiated Lomax distribution. Bayes point estimates of the model parameters and credible intervals under uniform and log-normal priors are obtained. Besides, optimum test plan based on constant stress ALT under Type I censoring is developed by minimizing the pre-posterior variance of a specified low percentile of the lifetime distribution at use condition. Gibbs sampling method is used to find the optimal stress with changing time. The performance of the estimation methods is demonstrated for both simulated and real data sets. Results indicate that both the priors and the sample size affect the optimal Bayesian plans. Further, informative priors provide better results than non-informative priors.

Optimal design of accelerated degradation test with multiple optimization objectives

ABSTRACT In this article, a multi-objective optimization method for the accelerated degradation test is proposed to solve the problem that the single objective optimization cannot meet the needs in practice gradually. Firstly, the Gamma process is assumed to describe the degradation path of product. Then, three optimization objectives are considered to establish the multi-objective optimization model based on the analytic hierarchy process, under the constraint that the total experimental cost does not exceed a predetermined budget. To solve the established optimization model, the self-adaption searching algorithm is proposed to search the optimal test plan, in which the searching interval of the decision variable is dynamically adjusted according to the feedback of the optimization function. Finally, the effectiveness of the proposed method is illustrated by a group of examples of motorized spindles.

Optimal durations of Weibull reliability tests based on failure counts

In industry, the durations of reliability experiments are usually fixed in advanced, which could lead to suboptimal results. A constrained optimization problem is stated in order to determine the best inspection time for reliability testing. The decision criterion is based on Weibull failure counts, whereas the constraints are related to the reliability and risk levels imposed by the producer and the consumer. The optimal reliability sampling plan provides the best strategy to determine the acceptability of lots and production processes. Minimum-cost inspection times, as well as the required number of test units and the maximum number of failures allowed, are derived by solving mixed integer nonlinear programming problems. An approximation of the minimum feasible acceptance number is first provided in closed-form. An efficient step-by-step procedure is then proposed in order to find the optimal reliability test plan for lot sentencing. In most cases, only a few iterations are needed to reach the optimal solution. For illustrative purposes, the suggested methodology is applied to the manufacturing of turbine engine combustors, systems of components and lawn mower motors.

Optimum degradation test plan under model mis-specification for Wiener and gamma processes

Degradation tests have been used for the purpose of assessing reliability information. In this paper, we consider a degradation test design problem under a statistical model mis-specification scenario. The Wiener and gamma processes are considered in this research. A gamma process is suitable for describing degradation paths that exhibit monotone behavior. However, if one fits a Wiener process model to the data, the resulting statistical inferences may be affected. Tsai, Tseng, and Balakrishnan studied the effect on the estimated mean time to failure. However, the experimental design problem was not discussed. A lack of the explicit functional form of the estimation variances makes it difficult to find degradation plans that can improve test efficiency. In this paper, functional forms of optimal degradation test plans are proposed under model mis-specification. Furthermore, a weighted ratio objective function considering the prior probability of the true model is used to find robust test plans for practical use. The results from a numerical example show that, by using an appropriate degradation test plan, the estimation variance can be reduced, and the test efficiency can be improved. A simulation study is conducted to investigate the performance of the parameter estimates when the sample size is small.

Optimal design of hybrid accelerated test based on the Inverse Gaussian process model

Accelerated life testing (ALT) and accelerated degradation testing (ADT) are two widely used accelerated testing (AT) methods for reliability evaluation of highly reliable products. However, in many cases, it is difficult to meet the estimation requirements only using ALT or ADT because of the nature of data and the test constraints. Since both ALT and ADT data provide useful reliability information, it would be quite valuable to use both of them for reliability estimation. In this paper, an optimal testing plan is proposed for the first time for such a hybrid AT under several experimental design constraints. An Inverse Gaussian (IG) process and the corresponding lifetime distribution are used to model the degradation process and the lifetime of the product, respectively. The strategy of allocating the test units to ALT and ADT and determining the data collection strategy in the experiment are presented. The objective is to maximize the estimation precision of the p-quantile of the product's lifetime under the normal use condition. A study on the stress relaxation of a type of electrical connector is conducted to illustrate the value of the proposed method in practice.

Optimal design of accelerated destructive degradation tests with block effects

Accelerated Destructive Degradation Tests (ADDTs) are effective for reliability assessment of highly reliable products whose key performance characteristic has to be destructively measured. Test units in a reliability experiment typically share the same test environments, and this introduces block effects to the resulting ADDT data. Nevertheless, the block effects are seldom considered in the optimal design of an ADDT plan. Motivated by an application of a seal strength test, this study discusses methods for planning ADDT with block effects. In particular, two types of block effects are considered, i.e., the rig-layer blocking due to a shared test rig, and the gauge-layer blocking resulting from simultaneous measurements. The ADDT planning specifies the optimal stress levels and allocation of test units to these stress levels to minimize the asymptotic variance of the estimated lifetime quantiles at use conditions. The optimal test plans are investigated analytically and through a comprehensive numerical study. An application to the motivating example reveals the importance of considering the block effects in the test design.

Batch sequential designs for accelerated life tests and an application to polymer composite fatigue test

Accelerated life tests (ALTs) are widely used in industry and many optimal ALT test plans are developed in literature. To achieve an efficient optimal test plan, one needs to make sure that the planning values used in the test planning are not far away from the true parameters, which is challenging because lifetime data are not collected yet in the planning stage to estimate the true parameters. To overcome the weakness, sequential designs are developed and the parameter estimates are updated during the sequential testing process. Due to the sequential nature, one drawback of the sequential strategy is that it usually requires a long total testing time (TTT). Motivated by industrial needs, we propose batch sequential design (BSD) in ALT setting, in which multiple units are tested simultaneously while sequential parameter updates are also made. The proposed BSD for ALT can not only reduce the TTT but also preserves the advantage of sequential designs. After proposing two types of BSD strategies, we use a comprehensive simulation study to demonstrate the advantages of the proposed BSD in ALT setting. We also develop a method to find the optimal batch size which is essential for the deployment of the BSD in practice.

Planning accelerated life tests with multiple sources of random effects

Accelerated life testing (ALT) is a commonly used experiment in industries for assessing a product’s lifetime. Planning a proper test with consideration of possible constraints on randomization is important because the prediction accuracy highly depends on the test plan. In this article, an optimal ALT test plan with multiple sources of random effects is demonstrated. Specifically, we consider two types of random effects caused by subsampling and blocking, and the experimental design consists of a crossed experimental factor – supplier, as well as a nested experimental factor – test chamber. The D-optimal ALT test plan is derived via a quasi-likelihood approach. An optimization algorithm with three iterative steps is developed to determine testing stress conditions, chamber assignments, and the number of test units for each testing condition and their supplier sources. The resulting test plans assign test units to chambers such that the effect of stress factor will not confound with the effect of test chamber and a similar number of test units from different suppliers will be required for each test condition.

Approximated optimal design for a bivariate step-stress accelerated life test with generalized exponential distribution under type-I progressive censoring

PurposeIn this paper, the author proposed an optimization design for a step-stress accelerated life test (SSALT) with two stress variables for the generalized exponential (GE) distribution under progressive type-I censoring.Design/methodology/approachIn this paper, two stress variables were considered. Progressive censoring and accelerated life testing were used to reduce the time and cost of testing. It was assumed that the lifetimes of the test units followed a GE distribution. The effects of changing stress were considered as a cumulative exposure model. A log-linear relationship between the scale parameter of the GE distribution and the stress was proposed. The maximum likelihood estimators and approximate and bootstrap confidence intervals (CIs) for the model parameters were obtained. An optimum test plan was developed using minimization of the asymptotic variance (AV) of the percentile life under the usual operating condition.FindingsAccording to the simulation results, the bootstrap CIs of the model parameters gave more accurate results than approximate CIs through the length of CIs. The sensitivity analysis was performed to illustrate the effect of initial estimates on optimal values that has been studied. Simulation results also indicated that the optimal times were not too sensitive to the initial values of parameters; thus, the proposed design was robust.Originality/valueIn most studies, only one accelerating stress variable is used. Sometimes accelerating one stress variable does not yield enough failure data. Thus, two stress variables may be needed for additional acceleration. In this paper, two stress variables are considered. The inclusion of two stress variables in a test design will lead to a better understanding of the effect of two simultaneously operating stress variables. Also, the author assumes that the failure time of the test units follows a GE distribution. It is observed that the GE distribution can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. Also, most studies in this field have focused on the derivation of optimum test plans. In this paper, the author examined the estimation of model parameters and the optimization of the test design. In this paper, the asymptotic and bootstrap CIs for the model parameters are calculated. In addition, a sensitivity analysis is performed to examine the effect of the changes in the pre-estimated parameters on the optimal hold times. For determining the optimal test plan, due to nonlinearity and complexity of the objective function, the particle swarm optimization (PSO) algorithm is developed to calculate the optimal hold times. In this method, the research speed is very fast and optimization ability is more.

Design of optimal sequential hybrid testing plans

One-shot units are produced in batches and stored in either a dormant or standby mode until retrieved or activated to perform their function when needed. In this article, we propose hybrid reliability testing approaches to utilize the advantages of non-destructive testing and destructive testing for assessment of reliability metrics. Specifically, we design a sequence of optimal hybrid testing plans under flexible scenarios. Extensive simulation models are developed to validate the accuracy and efficiency of the proposed approaches.

Explicit lot inspection by attributes using minimal prior information

An explicit optimal attribute sampling plan for lot acceptance with maximum allowable expected risks is presented without specifying the full prior model of the defective rate. The suggested methodology to find an approximate solution of the underlying integer nonlinear programming problem is simple, fast and sufficiently accurate in most practical cases. The smallest number of units to be inspected per lot in order to reach the required protections for customers and manufacturers, as well as the maximum tolerable number of defectives in the selected sample, is derived in closed-form using Taylor series expansions of the operating characteristic function. The proposed approach allows the practitioners to determine a nearly optimal inspection scheme using only prior estimations of the means and variances of the defective rates for the acceptable and rejectable lots. Furthermore, it provides easy ways to adequately combine multiple expert opinions and to update the current optimal test plan when new information becomes available. The incorporation of prior knowledge yields considerable savings in sample size, as well as improved evaluations of the current expected producer and consumer risks. Some applications in industrial reliability and quality control are included for illustrative and comparative purposes.

Optimal design of accelerated degradation test based on the combination forecast method for products with uncertain degradation models

The optimal design of an accelerated degradation test can effectively shorten the test duration and improve the accuracy of lifetime evaluation. This article proposes a new optimization method that considers the uncertainty of degradation models during the optimal design, and with the application of the combination forecast method based on the Akaike information criterion, establishes the optimization objective. Furthermore, the total test time is limited, with the inspection interval and the number of inspections as the design variables. A numerical example is presented to verify the effectiveness of the proposed method. The results show that when the degradation models of products are uncertain in the optimal design, the proposed method can be effectively used to obtain the optimum test plan; this makes an important contribution to reliability tests and assessment. Finally, a sensitivity analysis is conducted to assess the robustness of the optimum plan on parameter variation.

Bayesian Methods for Planning Accelerated Repeated Measures Degradation Tests

Accelerated repeated measures degradation tests are often used to assess product or component reliability when there would be few or even no failures during a traditional life test. Such tests are used to estimate the failure-time distributions of highly reliable items in applications where it is possible to take repeated measures of some appropriate degradation measure. When engineers have valid prior information about failure mechanisms, it is important that such information be used in inference and test planning. Bayesian methods provide a vehicle for doing this. This article describes methods for selecting a Bayesian repeated measures accelerated degradation test plan when the degradation and acceleration model is linear in the parameters. A Bayesian criterion based on estimation precision of the failure-time quantile at use conditions is selected for finding optimum test plans. We use a large-sample approximation for the posterior distribution to simplify the planning criterion. The general equivalence theorem is used to check for global optimality of the optimum test plan. We also discuss how to find a compromise test plan that satisfies practical constraints while still providing good statistical properties.

Construction of optimal reliability test plans for binary type multi-state strongly coherent systems

The concept of Binary Type Multi-State Strongly Coherent System (BTMSCS) or Multi-State Coherent System of Type 2 (MCS of type 2) has been defined by Natvig (1982). In this research article component based optimal reliability test plans for BTMSCS have been proposed. Here it is assumed that these systems are made up of n components, and the random variables representing lifetimes of systems and their components take values in set The construction of reliability test plans for BTMSCS is based on the use of (i) multivariate counting processes with marginal counting processes being homogeneous Poisson processes and (ii) positively upper orthant dependent property of the associated renewal processes. The applicability of the proposed reliability test plans has been demonstrated using all the three BTMSCSs made up of two components and one system consisting of three components with under the assumption that a bivariate random vector representing times spent by the ith component in states 1 and 2, follows Farlie-Gumbel-Morgenstern distribution with exponential marginals.

Optimal attribute sampling plans in closed-forms

Lot sampling inspection based on the number of defective units observed in selected random samples is widely used in reliability and quality control. An integer nonlinear programming problem is formulated to determine the most powerful binomial sampling plan for the disposition of large lots with limited producer and consumer risks. An appealing approximation of the optimal test plan is then explicitly deduced. The minimum number of units to be tested per lot in order to achieve the desired protections for manufacturers and customers, as well as the best decision rule for lot sentencing, is derived in closed-form. The procedure is simple, quick and accurate in most practical cases, and clearly improves the classical normal approximation. The suggested perspective allows the practitioners to greatly simplify the determination of the best attribute inspection scheme. In addition, it can be used to find the optimal reliability demonstration test plan based on the number of observed failures in a given time period. The developed methodology is applied to some examples in industrial manufacturing for illustrative and comparative purposes. A useful approximation of the optimal attribute sampling plan with given expected risks is also provided.

Optimum life test plan for Type-I hybrid censored Weibull distributed products sold under general rebate warranty

The selection of an appropriate life test plan is extremely important for any product as it not just improves the quality of the product but also reduces testing costs. In this approach, however, the choice of suitable costs play an important role. In this paper, a decision model is developed to determine the optimal life testing plan (LTP) by minimising the relevant costs involved for non-repairable products sold under the general rebate warranty. LTP are developed in the presence of Type-I hybrid censoring for products having Weibull distributed lifetimes. A constrained optimisation approach is followed considering both producer's and consumer's risk and suitable analysis techniques are employed in obtaining the optimal solution. Monte Carlo simulation is conducted for numerical illustration. In order to study the sensitivity of the optimal solution due to mis-specification of parameter values and cost components, a well-designed sensitivity analysis is incorporated using a real-life Type-I hybrid censored data set.

Robustness of asymptotic accelerated life test plans to small‐sample settings

AbstractMuch of the statistical literature on optimal test planning for accelerated life testing utilize asymptotic methods to derive optimal test plans. While sufficient effort is made to assess the robustness of these test plans to the choice of design parameters and distribution assumptions, there is very little literature on the performance of asymptotic test plans relative to small samples (on the order of 10‐15 samples). An alternative concern is that the asymptotic test plans may not necessarily be the true “optimal” test plan for a given sample size. The purpose of this research is to present exact or “near‐exact” methods for developing test plans and compare the performance of these test plans with corresponding asymptotic test plans in small‐sample settings. The optimal location of design points and sample allocation is determined using each method for lognormal and Weibull lifetime distributions with both complete and Type 1 right‐censored data under two selected acceleration factor models. The investigations reveal that asymptotic test plans tend to corroborate quite well with exact test plans and thus are suitably robust to small‐sample settings in terms of optimal variance.

Planning accelerated reliability tests for mission-oriented systems subject to degradation and shocks

This article presents a novel accelerated reliability testing framework for mission-oriented systems. The system to be tested is assumed to suffer from cumulative degradation and traumatic shocks with increasing intensity. We propose a new optimality criterion that minimizes the asymptotic variance of the predicted reliability evaluated at the mission’s end time. Two usage scenarios are considered in this study: one is to assume that systems are brand new at the start of the mission and the other is that systems are randomly selected from used ones under pre-determined policies. Optimal test plans for both scenarios are obtained via delta methods by utilizing the Fisher information. The global optimality of test plans is verified using general equivalence theorems. A revisited example of a carbon-film resistor is presented to illustrate the efficiency and robustness of optimal test plans for both new and randomly aged systems. The result shows that the test plan tends to explore more on lower stress levels for randomly aged systems. Furthermore, we conduct simulation studies and explore compromise test plans for the example.