Weibull Lifetime Distribution Research Articles

Reliability estimation in a Weibull lifetime distribution with zero‐failure field data

AbstractThe estimation of product reliability has attracted worldwide attention during the past several decades. The estimation procedure usually begins with parameter estimation based on test data. When there is no failure occurring in tests, traditional approaches like Maximum Likelihood Estimation (MLE) cannot be applied to estimate parameters. When product lifetime follows a Weibull distribution, to cope with this problem, we propose the modified MLE (MMLE) for estimating the parameters, based on the zero‐failure data. In this paper, we also consider the prior reliability estimate from a similar product and make use of it by incorporating it with the MMLE to construct the shrinkage preliminary test estimator (SPTE). We present the calculation method of the shrinkage factor in the SPTE, by referring to the comparison of critical quality characteristics related to product reliability, between the current batch of products and the similar (or earlier version) batch of products. Restrictions for the shrinkage factor to ensure the performance of SPTE are also discussed. The example demonstrates that the proposed SPTE of the product reliability is an effective methodology to estimate the product reliability and improve the estimation performance of the MMLE, when only zero‐failure data are available. Copyright © 2010 John Wiley & Sons, Ltd.

Preventive-maintenance policy for leased products under various maintenance costs

This paper investigates the effects of preventive-maintenance cost functions on the optimal preventive-maintenance policy for a leased product with Weibull life-time distribution. During the lease period, any product failures are rectified by minimal repairs and may incur a penalty to the lessor, if the time duration for performing a minimal repair exceeds a pre-specified time limit. To reduce repair costs and possible penalty, preventive-maintenance actions are scheduled in the lease contract. The objective of this paper is to derive the optimal preventive-maintenance schedule and maintenance degrees such that the expected total cost is minimized. Some structural properties of the optimal policy are obtained and an efficient algorithm is provided to search for the optimal policy. For the cases where the preventive maintenance cost is a constant or a linearly increasing function, the effects of the preventive-maintenance cost function on the optimal policy are investigated in detail both theoretically and numerically.

Reliability acceptance sampling plans for the Weibull distribution under accelerated Type-I censoring

Type-I censored reliability acceptance sampling plans (RASPs) are developed for the Weibull lifetime distribution with unknown shape and scale parameters such that the producer and consumer risks are satisfied. It is assumed that the life test is conducted at an accelerated condition for which the acceleration factor (AF) is known, and each item is continuously monitored for failure. Sensitivity analyses are also conducted to assess the effect of the uncertainty in the assumed AF on the actual producer and consumer risks, and a method is developed for constructing RASPs that can accommodate the uncertainty in AF.

On the estimation of reliabilty function in a Weibull lifetime distribution

The estimation of the reliability function of the Weibull lifetime model is considered in the presence of uncertain prior information (not in the form of prior distribution) on the parameter of interest. This information is assumed to be available in some sort of a realistic conjecture. In this article, we focus on how to combine sample and non-sample information together in order to achieve improved estimation performance. Three classes of point estimatiors, namely, the unrestricted estimator, the shrinkage estimator and shrinkage preliminary test estimator (SPTE) are proposed. Their asymptotic biases and mean-squared errors are derived and compared. The relative dominance picture of the estimators is presented. Interestingly, the proposed SPTE dominates the unrestricted estimator in a range that is wider than that of the usual preliminary test estimator. A small-scale simulation experiment is used to examine the small sample properties of the proposed estimators. Our simulation investigations have provided strong evidence that corroborates with asymptotic theory. The suggested estimation methods are applied to a published data set to illustrate the performance of the estimators in a real-life situation.

Eliciting vague but proper maximal entropy priors in Bayesian experiments

Priors elicited according to maximal entropy rules have been used for years in objective and subjective Bayesian analysis. However, when the prior knowledge remains fuzzy or dubious, they often suffer from impropriety which can make them uncomfortable to use. In this article we suggest the formal elicitation of an encompassing family for the standard maximal entropy (ME) priors and the maximal data information (MDI) priors, which can lead to obtain proper families. An interpretation is given in the objective framework of channel coding. In a subjective framework, the performance of the method is shown in a reliability context when flat but proper priors are elicited for the Weibull lifetime distributions. Such priors appear as practical tools for sensitivity studies.

Acceptance sampling plans under progressive interval censoring with likelihood ratio

The paper investigates the design of life test plans under progressively interval censored test. Based on the likelihood ratio, the proposed life test plans are established so that the required producer and consumer risks can be satisfied simultaneously. The advantage of the proposed method is that the developed sampling procedure depends on the likelihood ratio only so that the method can be applied to any lifetime distribution when only one parameter is unknown. A numerical study is conducted and some of the sampling plans for the Weibull lifetime distribution with different shape parameters are tabulated for illustration. Moreover, the influence of the removal schemes on the proposed sampling plans is discussed.

Reliability sampling plans for Weibull distribution with limited capacity of test facility

This paper establishes reliability sampling plans for the Weibull lifetime distribution based on type II censored data with limited capacity of test facility. The products are sold under a general rebate warranty policy. It is also assumed that the shape parameter of the lifetime distribution is known, and the scale parameter is a random variable varying from lot to lot. A cost model is established which contains the cost per unit on test, the cost per unit time for life test, and the costs of rejecting and accepting a unit. An algorithm for determining the optimal reliability sampling plans which minimize the expected average cost per lot is provided. Some numerical results to illustrate the use of the proposed method are studied.

Optimal threshold value of failure-rate for leased products with preventive maintenance actions

This paper investigates the optimal threshold value of failure rates for leased products with a Weibull lifetime distribution. Within a lease period, any product failure is rectified by minimal repairs and the lessor may incur a penalty when the time required to perform a minimal repair exceeds a reasonable time limit. To reduce product failures, additional preventive maintenance actions are carried out when the failure rate reaches a threshold value. Under this maintenance scheme, a mathematical model of the expected total cost is established. Based on the model, the optimal threshold value and the corresponding maintenance degrees are derived such that the expected total cost is minimized. The structural properties of the optimal policy are investigated and an efficient algorithm is provided to search for the optimal policy. Finally, numerical examples are given to illustrate the features of the optimal policy.

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This paper presents the reliability estimation of door hinge for home appliances, which consists of bushing and shaft. The predominant failure mechanism of bushing made of polyoxymethylene(POM) is brittle fracture due to decrease of strength caused by voids existing, and that of shaft made of acrylonitrile-butadiene-styrene(ABS) is creep due to plastic deformation caused by excessive temperature and lowering of glass transition temperature by absorbed moisture. Since the brittle fracture of bushing is overstress failure mechanism, the load-strength interference model is used to estimate the failure rate of it along with failure analysis. By the way, the creep of shaft is wear out failure mechanism, and an accelerated life test is then planned and implemented to estimate its lifetime. Through the technical review about failure mechanism, temperature and humidity are selected as accelerating variables. Assuming Weibull lifetime distribution and Eyring model, the life-stress relationship and acceleration factor, BIO life and its lower bound with 90% confidence at worst case use condition are estimated by analyzing the accelerated life test data.

Competing Causes of Failure and Reliability Tests for Weibull Lifetimes Under Type I Progressive Censoring

For many high reliability products where very few items are expected to fail during the test period, testing under normal conditions is not feasible. Further, the requirement for high reliability increases the need for test procedures which yield valuable degradation and other useful information for improving product reliability. Thus in some manufacturing and other experiments, various types of failure censored and accelerated life tests are commonly employed for life testing. In this paper we discuss Type I progressively censored variable-sampling plans for Weibull lifetime distributions under competing causes of failure. The proposed procedure is attractive as it yields useful degradation-related information for improving product quality. In addition, the procedure is useful when a test is conducted under severe time constraint and/or when the experimenter wishes to save costly specimens or scarce test facilities for other use.

Reliability Estimation and Failure Analysis of Multilayer Ceramic Chip Capacitors

This paper presents the failure analysis and the reliability estimation of a multilayer ceramic chip capacitor. For the failed samples used in an automobile engine control unit, failure analysis was made to identify the root cause of failure and it was shown that the migration and the avalanche breakdown were the dominant failure mechanisms. Next, an accelerated life testing was designed to estimate the life of the MLCC. It is assumed that Weibull lifetime distribution and the life-stress relationship proposed Prokopowicz and Vaskas. The life-stress relationship and the acceleration factor are estimated by analyzing the accelerated life test data.

Reliability Evaluation and Failure Analysis for NTC Thermistor

Failures of NTC thermistor are analyzed. Visual inspection, electrical parameter test, non-destructive test, and destructive physical analysis were performed on the field samples to identify the root cause of failure. It was found that the predominant failure mechanism was the copper migration of dumet (copper-clad alloy) at glass-PVC interface for glass-coated chip type NTC thermistor molded with PVC. Next, an accelerated life test is designed to predict the lifetime. Temperature, voltage, and humidity are considered as accelerating variables. Under the assumptions of a general life-stress relationship and Weibull lifetime distribution, the parameters of life-stress relationship and acceleration factor for the migration are estimated by analyzing the test data.

Accelerated Life Test for the Embrittlement of Natural Rubber Grommets

This paper presents an accelerated life test for embrittlement of natural rubber grommets. From the analyses of field samples, it is found that embrittlement is due to the decrease of plasticizer (zinc stearate for natural rubber) in the rubber. To estimate the embrittlement life, an accelerated life test is designed based on a factorial design. Temperature and compression load are selected as accelerating variables, and the hardness is measured for each specimen during the test. An analysis of variance indicates that the significant factor affecting the hardness is not load, but temperature. The embrittlement life and temperature relationship and the acceleration factor are estimated from the failure times using linear degradation model under the assumptions of Weibull lifetime distribution and Arrhenius model.

Statistical Aspects of Lifetime in the Presence of UV Radiation

The reliability function of a component cannot be satisfactorily estimated from experiments, because: (i) an accurate estimation of the lifetime distribution tails, controlling the most important domain of high reliability, requires a very large sample and (ii) reliability tests under normal operational conditions are necessarily very lengthy. Hence the urgent need for a physical model for component lifetime statistics. The paper presents an application of the recently developed model for damage accumulation in polymeric materials to the long-term constant stress rupture experiments on Kevlar Composite, which is widely used in fiber optics. The strong dependence of the experimentally observed distribution shape on the load applied to a component has been previously explained in the framework of two different damage mechanisms: kinetic crack growth and chemical deterioration; the resultant 3-parameter lifetime distribution was predicted to be essentially non-Weibull. The proposed model, based on a single micro-mechanical damage mechanism, leads to a 2-parameter Weibull lifetime distribution with the shape parameter depending on the applied load by a simple inverse power law. Both distribution models were fitted to experimental lifetime data for different stress levels and the corresponding goodness of fit was compared by the usual likelihood ratio test. The proposed model describes the experimental data better — especially in the most important domain of low stress and long (of the order of years) lifetime. The model is physically sound and permits improved design of the accelerated tests and more accurate interpretation of their results, and finally quantitative prediction of the reliability function of the loaded polymeric component.

Estimation of parameters from progressively censored data using EM algorithm

EM algorithm is used to determine the maximum likelihood estimates when the data are progressively Type II censored. The method is shown to be feasible and easy to implement. The asymptotic variances and covariances of the ML estimates are computed by means of the missing information principle. The methodology is illustrated with two popular models in lifetime analysis, the lognormal and Weibull lifetime distributions.

Performance assessment of multiple engineered barrier systems

The lifetime of the engineered barrier system used for containment of high-level radioactive waste will significantly impact the total performance of a geological repository facility. Currently two types of designs are under consideration for an engineered barrier system, single engineered barrier system and multiple engineered barrier system. Multiple engineered barrier system consists of several metal barriers and the waste form (cladding). Some recent work showed that a significant improvement of performance can be achieved by utilizing multiple engineered barrier systems. Considering sequential failures for each barrier, we model the reliability of the multiple engineered barrier system. Weibull and exponential lifetime distributions are used through out the analysis. Furthermore, the number of failed engineered barrier systems in a repository at a given time is modeled using a Poisson approximation.

A stochastic regime switching model for the failure process of a repairable system

This paper presents a stochastic model and estimation procedure for analyzing the failure process of a repairable system. We consider repairable systems whose successive interfailure times reveal a significant dependence while showing an insignificant trend. Neither the renewal process nor the non-homogeneous Poisson process are adequate for modeling such failure processes. Especially when the interfailure times show a cyclic pattern, we may consider a switching of the regimes (states) governing the lifetime distribution of the system. We propose a Markov switching model describing the failure process for such a case. The model postulates that a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain. Each of the distinct lifetime distributions represents a failure type that may change after successive repairs. Our model generalizes the mixture model by allowing the mixture probabilities to change during the transient period of the system. The model can capture the transient behavior of the system. The interfailure times constitute a set of incomplete data because the states are not explicitly identified. For the incomplete data, we propose a procedure for finding the maximum likelihood estimates of the model parameters by adopting the expectation and maximization principle. We also suggest a statistical method to determine the number of significant states. A Monte Carlo study is performed with two-parameter Weibull lifetime distributions. The results show consistency and good properties of the estimates. Some sets of Proschan's air conditioning unit data [ Technometrics, 1963, 5′ 375–383] are also analyzed and the results are discussed with respect to the number of significant states and the performance of the prediction.

A Bayesian life test sampling plan for products with Weibull lifetime distribution sold under warranty

A Bayesian life test sampling plan is considered for products with Weibull lifetime distribution which are sold under a warranty policy. It is assumed that the shape parameter of the distribution is a known constant, but the scale parameter is a random variable varying from lot to lot according to a known prior distribution. A cost model is constructed which involves three cost components; test cost, accept cost, and reject cost. A method of finding optimal sampling plans which minimize the expected average cost per lot is presented and sensitivity analyses for the parameters of the lifetime and prior distributions are performed.

Reliability modeling of engineered barrier systems for nuclear waste: A conditional approach

The lifetime of the Engineered Barrier System (EBS) used for containment of high-level radioactive waste will play an important role in controlling the cumulative release of radionuclides from a geological repository. Essentially, an EBS has two major components: a metal barrier and the waste form (cladding). Since any suitable site is above the static water table, the decay heat generated from this waste is expected to keep the surface temperature of a canister above the boiling point of water for a T random time period. Assuming the Weibull lifetime distribution on each component, we use the conditional approach to derive the lifetime distribution of the EBS. In our model, the oxidization process of the metal barrier (i.e. the lifetime of the metal barrier) starts after this time T and the lifetime of the cladding starts after the lifetime of the metal barrier

Weibull renewal processes

We study the class of renewal processes with Weibull lifetime distribution from the point of view of the general theory of point processes. We investigate whether a Weibull renewal process can be expressed as a Cox process. It is shown that a Weibull renewal process is a Cox process if and only if 0<α≤1, where α denotes the shape parameter of the Weibull distribution. The Cox character of the process is analyzed. It is shown that the directing measure of the process is continuous and singular.