Designing Adaptive Multiple Dependent State Sampling Plan for Accelerated Life Tests

A novel adaptive multiple dependent state sampling plan(AMDSSP)was designed to inspect products from a continuous manufacturing process under the accelerated life test(ALT)using both double sampling plan(DSP)and multiple dependent state sampling plan(MDSSP)concepts.Under accelerated conditions,the lifetime of a product follows the Weibull distribution with a known shape parameter,while the scale parameter can be determined using the acceleration factor(AF).The Arrhenius model is used to estimate AF when the damaging process is temperature-sensitive.An economic design of the proposed sampling plan was also considered for the ALT.A genetic algorithm with nonlinear optimization was used to estimate optimal plan parameters to minimize the average sample number(ASN)and total cost of inspection(TC)under both producer’s and consumer’s risks.Numerical results are presented to support the AMDSSP for the ALT,while performance comparisons between the AMDSSP,the MDSSP and a single sampling plan(SSP)for the ALT are discussed.Results indicated that the AMDSSP was more flexible and efficient for ASN and TC than the MDSSP and SSP plans under accelerated conditions.The AMDSSP also had a higher operating characteristic(OC)curve than both the existing sampling plans.Two real datasets of electronic devices for the ALT at high temperatures demonstrated the practicality and usefulness of the proposed sampling plan.

Analysis of Incomplete Data of Accelerated Life Testing with Competing Failure Modes

Data obtained from accelerated life testing （ALT） when there are two or more failure modes, which is commonly referred to as competing failure modes, are often incomplete. The incompleteness is mainly due to censoring, as well as masking which might be the case that the failure time is observed, but its corresponding failure mode is not identified. Because the identification of the failure mode may be expensive, or very difficult to investigate due to lack of appropriate diagnostics. A method is proposed for analyzing incomplete data of constant stress ALT with competing failure modes. It is assumed that failure modes have s-independent latent lifetimes and the log lifetime of each failure mode can be written as a linear function of stress. The parameters of the model are estimated by using the expectation maximum （EM） algorithm with incomplete data. Simulation studies are performed to check＇model validity and investigate the properties of estimates. For further validation, the method is also illustrated by an example, which shows the process of analyze incomplete data from ALT of some insulation system. Because of considering the incompleteness of data in modeling and making use of the EM algorithm in estimating, the method becomes more flexible in ALT analysis.

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.

Estimation for constant-stress accelerated life test from generalized half-normal distribution

In the constant-stress accelerated life test, estimation issues are discussed for a generalized half-normal distribution under a log-linear life-stress model. The maximum likelihood estimates with the corresponding fixed point type iterative algorithm for unknown parameters are presented, and the least square estimates of the parameters are also proposed. Meanwhile, confidence intervals of model parameters are constructed by using the asymptotic theory and bootstrap technique. Numerical illustration is given to investigate the performance of our methods.

Optimum design of equivalent accelerated life testing plans based on proportional hazards-proportional odds model

The optimum design of equivalent accelerated life testing plan based on proportional hazards-proportional odds model using D-optimality is presented. The defined equivalent test plan is the test plan that has the same value of the determinant of Fisher information matrix. The equivalent test plan of step stress accelerated life testing （SSALT） to a baseline optimum constant stress accelerated life testing （CSALT） plan is obtained by adjusting the censoring time of SSALT and solving the optimization problem for each case to achieve the same value of the determinant of Fisher information matrix as in the baseline optimum CSALT plan. Numer- ical examples are given finally which demonstrate the equivalent SSALT plan to the baseline optimum CSALT plan reduces almost half of the test time while achieving approximately the same estimation errors of model parameters.

Design of Accelerated Life Test Plans——Overview and Prospect

Accelerated life test(ALT) is currently the main method of assessing product reliability rapidly, and the design of efficient test plans is a critical step to ensure that ALTs can assess the product reliability accurately, quickly, and economically. With the promotion of the national strategy of civil-military integration, ALT will be widely used in the research and development(R&D) of various types of products, and the ALT plan design theory will face further challenges. To aid engineers in selecting appropriate theories and to stimulate researchers to develop the theories required in engineering, with focus on the demands for theory research that arise from the implementation of ALT, this paper reviews and summarizes the development of ALT plan design theory. The development of the theory and method for planning optimal ALT for location-scale distribution, which is the most applied and mature theory of designing the optimal ALT plan, are described in detail. Taking this as the center of radiation, some problems that ALT now faces, such as the verification of the statistical model, limitation of sample size, solutions of resource limits, optimization of the test arrangement, and management of product complexity, are discussed, and the general ideas and methods of solving these problems are analyzed. Suggestions for selecting appropriate ALT plan design theories are proposed, and the urgent solved theory problems and opinions of their solutions are proposed. Based on the principle of convenience for engineers to select appropriate methods according to the problems found in practice, this paper reviews the development of optimal ALT plan design theory by taking the engineering problems arising from the ALT implementation as the main thread, provides guidelines on selecting appropriate theories for engineers, and proposes opinions about the urgent solved theory problems for researchers.

Parameter Estimation Based on Censored Data under Partially Accelerated Life Testing for Hybrid Systems due to Unknown Failure Causes

In general,simple subsystems like series or parallel are integrated to produce a complex hybrid system.The reliability of a system is determined by the reliability of its constituent components.It is often extremely difficult or impossible to get specific information about the component that caused the system to fail.Unknown failure causes are instances in which the actual cause of systemfailure is unknown.On the other side,thanks to current advanced technology based on computers,automation,and simulation,products have become incredibly dependable and trustworthy,and as a result,obtaining failure data for testing such exceptionally reliable items have become a very costly and time-consuming procedure.Therefore,because of its capacity to produce rapid and adequate failure data in a short period of time,accelerated life testing(ALT)is the most utilized approach in the field of product reliability and life testing.Based on progressively hybrid censored(PrHC)data froma three-component parallel series hybrid system that failed to owe to unknown causes,this paper investigates a challenging problem of parameter estimation and reliability assessment under a step stress partially accelerated life-test(SSPALT).Failures of components are considered to follow a power linear hazard rate(PLHR),which can be used when the failure rate displays linear,decreasing,increasing or bathtub failure patterns.The Tempered random variable(TRV)model is considered to reflect the effect of the high stress level used to induce early failure data.The maximum likelihood estimation(MLE)approach is used to estimate the parameters of the PLHR distribution and the acceleration factor.A variance covariance matrix(VCM)is then obtained to construct the approximate confidence intervals(ACIs).In addition,studentized bootstrap confidence intervals(ST-B CIs)are also constructed and compared with ACIs in terms of their respective interval lengths(ILs).Moreover,a simulation study is conducted to demonstrate the performance of the estimation procedures and the methodology discussed in this paper.Finally,real failure data from the air conditioning systems of an airplane is used to illustrate further the performance of the suggested estimation technique.

Inference for constant-stress accelerated life test with Type-I progressively hybrid censored data from Burr-XII distribution

This paper proposes a simple constant-stress accel- erated life test （ALT） model from Burr type XII distribution when the data are Type-I progressively hybrid censored. The maximum likelihood estimation （MLE） of the parameters is obtained through the numerical method for solving the likelihood equations. Approxi- mate confidence interval （CI）, based on normal approximation to the asymptotic distribution of MLE and percentile bootstrap Cl is derived. Finally, a numerical example is introduced and then a Monte Carlo simulation study is carried out to illustrate the pro- posed method.

Inference and optimal design on step-stress partially accelerated life test for hybrid system with masked data

Under Type-Ⅱ progressively hybrid censoring, this paper discusses statistical inference and optimal design on stepstress partially accelerated life test for hybrid system in presence of masked data. It is assumed that the lifetime of the component in hybrid systems follows independent and identical modified Weibull distributions. The maximum likelihood estimations(MLEs)of the unknown parameters, acceleration factor and reliability indexes are derived by using the Newton-Raphson algorithm. The asymptotic variance-covariance matrix and the approximate confidence intervals are obtained based on normal approximation to the asymptotic distribution of MLEs of model parameters. Moreover,two bootstrap confidence intervals are constructed by using the parametric bootstrap method. The optimal time of changing stress levels is determined under D-optimality and A-optimality criteria.Finally, the Monte Carlo simulation study is carried out to illustrate the proposed procedures.

Bayesian Analysis in Partially Accelerated Life Tests for Weighted Lomax Distribution

Accelerated life testing has been widely used in product life testing experiments because it can quickly provide information on the lifetime distributions by testing products or materials at higher than basic conditional levels of stress,such as pressure,temperature,vibration,voltage,or load to induce early failures.In this paper,a step stress partially accelerated life test(SSPALT)is regarded under the progressive type-II censored data with random removals.The removals from the test are considered to have the binomial distribution.The life times of the testing items are assumed to follow lengthbiased weighted Lomax distribution.The maximum likelihood method is used for estimating the model parameters of length-biased weighted Lomax.The asymptotic confidence interval estimates of the model parameters are evaluated using the Fisher information matrix.The Bayesian estimators cannot be obtained in the explicit form,so the Markov chain Monte Carlo method is employed to address this problem,which ensures both obtaining the Bayesian estimates as well as constructing the credible interval of the involved parameters.The precision of the Bayesian estimates and the maximum likelihood estimates are compared by simulations.In addition,to compare the performance of the considered confidence intervals for different parameter values and sample sizes.The Bootstrap confidence intervals give more accurate results than the approximate confidence intervals since the lengths of the former are less than the lengths of latter,for different sample sizes,observed failures,and censoring schemes,in most cases.Also,the percentile Bootstrap confidence intervals give more accurate results than Bootstrap-t since the lengths of the former are less than the lengths of latter for different sample sizes,observed failures,and censoring schemes,in most cases.Further performance comparison is conducted by the experiments with real data.

Empirical Approach for Planning and Designing Constant Stress Accelerated Life Tests

Accelerated life testing(ALT)has been widely used to obtain information about the product's life characteristics at normal conditions in a relatively short period of time.Two key issues with ALT are test design and data analysis.The test design of constant stress ALT was studied in this paper.The test design usually combines engineering experiences with optimization models.Such approaches are hard to be implemented by practitioners.A"pure"empirical approach was presented to address this issue.With the proposed approach,some of the decision variables are determined based on the results from the literature,some of the other variables are determined based on engineering analysis and /or judgment,and the remaining variables are determined based on the empirical relations developed in this paper.A real-world example is included to illustrate the appropriateness of the proposed approach.

Constant-Step Stress Accelerated Life Test of VFD under Logarithmic Normal Distribution Case

In order to solve the life problem of vacuum fluorescent display (VFD) within shorter time, and reduce the life prediction cost, a constant-step stress accelerated life test was performed with its cathode temperature increased. Statistical analysis was done by applying logarithmic normal distribution for describing the life, and least square method (LSM) for estimating logarithmic normal parameters. Self-designed special software was used to predict the VFD life. It is verified by numerical results that the VFD life follows logarithmic normal distribution, and that the life-stress relationship satisfies linear Arrhenius equation completely. The accurate calculation of the key parameters enables the rapid estimation of VFD life.

Constant Stress Accelerated Life Tests for Vacuum Fluorescent Display Based on Least Square Method

To estimate the life of vacuum fluorescent display （VFD） more accurately and reduce test time and cost, four constant stress accelerated life tests （CSALTs） were conducted on an accelerated life test model. In the model, statistical analysis of test data is achieved by applying lognormal function to describe the life distribution, and least square method （LSM） to calculate the mean value and the standard deviation of logarithm. As a result, the accelerated life equation was obtained, and then a self-developed software was developed to predict the VFD life. The data analysis results demonstrate that the VFD life submits to lognormal distribution, that the accelerated model meets the linear Arrhenius equation, and that the precise accelerated parameter makes it possible to acquire the life information of VFD within one month.

Capacity Configuration of Energy Storage Systems for Echelon Utilization Based on Accelerated Life Test in Microgrids

Retired power battery construction energy storage systems(ESSs)for echelon utilization can not only extend the remaining capacity value of the battery,and decrease environmental pollution,but also reduce the initial cost of energy storage systems.In this paper,an ESS constructed of retired power batteries for echelon utilization in microgrids(MGs)is considered.Firstly,considering the influence of different discharge depths on the battery life cycle,the correlation equation between the state of charge(SOC)and the state of health(SOH)is established.Secondly,the accelerated life test method,based on the inverse power law coefficient equation,is proposed,and it is used to evaluate the reliability of the ESS.Finally,according to the SOC characteristics,the dynamic security margin of the ESS is established.The life cycle cost,supply-demand balance and ESS balanced control are comprehensively considered,and the location and capacity of energy storage in MGs are determined.It is simulated using the IEEE-RTS 24 node system;the results show that the investment cost of the ESS is reduced and the operational life is prolonged.

An accelerated life test model for harmonic drives under a segmental stress history and its parameter optimization

Harmonic drives have various distinctive advantages and are widely used in space drive mechanisms. Accelerated life test （ALT） is commonly conducted to shorten test time and reduce associated costs. An appropriate ALT modet is needed to predict the lifetime of harmonic drives with ALT data. However, harmonic drives which are used in space usually work under a segmental stress history, and traditional ALT models can hardly be used in this situation. This paper proposes a dedicated ALT model for harmonic drives applied in space systems. A comprehensive ALT model is established and genetic algorithm （GA） is adopted to obtain optimal parameters in the model using the Manson fatigue damage rule to describe the fatigue failure process and a cumulative dam- age method to calculate and accumulate the damage caused by each segment in the stress history. An ALT of harmonic drives was carried out and experimental results show that this model is acceptable and effective.

Bayesian Planning of Optimal Step-stress Accelerated Life Test for Log-location-scale Distributions

This paper introduces some Bayesian optimal design methods for step-stress accelerated life test planning with one accelerating variable, when the acceleration model is linear in the accelerated variable or its function, based on censored data from a log-location-scale distributions. In order to find the optimal plan,we propose different Monte Carlo simulation algorithms for different Bayesian optimal criteria. We present an example using the lognormal life distribution with Type-I censoring to illustrate the different Bayesian methods and to examine the effects of the prior distribution and sample size. By comparing the different Bayesian methods we suggest that when the data have large（small） sample size B1（τ）（B2（τ）） method is adopted. Finally, the Bayesian optimal plans are compared with the plan obtained by maximum likelihood method.

Analysis and Optimum Plan for 3-Step Step-Stress Accelerated Life Tests with Lomax Model Under Progressive Type-I Censoring

Inthispaper,theoptimumtestplanandparameterestimationfor3-stepstep-stress accelerated life tests in the presence of modified progressive Type-I censoring are discussed.It is assumed that the lifetime of test units follows a Lomax distribution with log of characteristic life being quadratic function of stress level.The maximum likelihood and Bayesian method are used to obtain the point and interval estimators of the model parameters.The Bayes estimates are obtained using Markov chain Monte Carlo simulation based on Gibbs sampling.The optimum plan for 3-step step-stress test under modified progressive Type-I censoring is developed which minimizes the asymptotic variance of the maximum likelihood estimators of log of scale parameter at design stress.Finally,the numerical study with sensitivity analysis is presented to illustrate the proposed study.

Comparison Analysis of Efficiency Between Double-Synchronous Step-Down-Stress and Step-Up-Stress Accelerated Life Tests

Double-crossed-step-stress(DCSS) accelerated life test(ALT) method is widely used for estimating the lifetime of products with high reliability and long lifetime. In order to further reduce the test time and test cost, a double-synchronous-step-stress(DSSS) ALT method which combines a double-synchronous-step-downstress(DSSDS) ALT method and a double-synchronous-step-up-stress(DSSUS) ALT method is proposed. The accelerated stresses decrease and increase in a synchronous way with one step in the DSSDS-ALT and DSSUSALT methods, respectively. Monte Carlo method is adopted to simulate the two methods, and the validity and efficiency of them are demonstrated by the simulation results. In addition, a comparison analysis of efficiency between DSSDS-ALT method and DSSUS-ALT method is carried out. The result shows that the DSSDS-ALT method compared with the DSSUS-ALT method can significantly improve the test efficiency under the same test condition.

Acceptance sampling plan of accelerated life testing for lognormal distribution under time-censoring

Abstract Lognormal distribution is commonly used in engineering. It is also a life distribution of important research values. For long-life products follow this distribution, it is necessary to apply accelerated testing techniques to product demonstration. This paper describes the development of accelerated life testing sampling plans （ALSPs） for lognormal distribution under time-censoring conditions. ALSPs take both producer and consumer risks into account, and they can be designed to work whether acceleration factor （AF） is known or unknown. When AF is known, lift testing is assumed to be conducted under accelerated conditions with time-censoring. The producer and con- sumer risks are satisfied, and the size of test sample and the size of acceptance number arc opti- mized. Then sensitivity analyses are conducted. When AF is unknown, two or more predetermined levels of accelerated stress are used. The sample sizes and sample proportion allo- cated to each stress level are optimized. The acceptance constant that satisfies producer and consumer risk is obtdned by minimizing the generalized asymptotic variance of the test statistics. Finally, the properties of the two ALSPs （one for known-AF conditions and one for unknown AF conditions） are investigated to show that the proposed method is corrcct and usablc through numerical examples.

Identifying the failure mechanism in accelerated life tests by two-parameter lognormal distributions

The failure mechanism stimulated by accelerated stress in the degradation may be different from that under normal conditions, which would lead to invalid accelerated life tests. To solve the problem, we study the re- lation between the Arrhenius equation and the lognormal distribution in the degradation process. Two relationships of the lognormal distribution parameters must be satisfied in the conclusion of the unaltered failure mechanism, the first is that the logarithmic standard deviations must be equivalent at different temperature levels, and the second is that the ratio of the differences between logarithmic means must be equal to the ratio of the differences between reciprocals of temperature. The logarithm of distribution lines must simultaneously have the same slope and regular interval lines. We studied the degradation of thick-film resistors in MCM by accelerated stress at four temperature levels （390, 400, 410 and 420 K）, and the result agreed well with our method.