Some contributions to optimal reliability test plans and estimation

Abstract

Design of optimal system reliability test plan has numerous advantages in product manufacturing industry as it helps to evaluate the system reliability and thereby demonstrates that the system will perform satisfactorily, prior to its deployment to the concerned eld. The failure data and prior information available on failure rates of units will help manufacturers to develop higher warranty periods and better ser- vice facilities to customers. Thus, a Bayesian approach can be adopted to obtain a good reliability estimates and cost-e ective optimal test plans. In systems that are designed to achieve very high reliability (e.g., missiles, rockets, etc.), it is di cult to obtain good estimate of system reliability. This is due to either the data recorded contain only a small number of failures or lack of availability of su cient testing time to observe failures. In such circumstances, a better alternative for obtaining data quickly is the application of accelerated life tests (ALTs) or partially accelerated life tests (PALTs). In both of these testing procedures, units are subjected to perform under higher stress than the normal stress level, whereas in PALT, some units are allowed to perform under normal stress. Both testing procedures are destructive in nature, i.e., one has to destroy some units to obtain lifetime quickly. Thus, in most of the existing reliability estimation method, failure time data are obtained from either classical life test or accelerated testing. In most of such situations, observing data is time-consuming, expensive or impracticable. One can make use of readily available degradation data enclosing information about system failure. For example, a uo- rescent lamp is considered as a failed one, when its luminosity falls below a certain level, where an interesting feature of this experiment is the periodic monitoring of the luminosity. Hence degradation data, which are obtained during the lifetime of iv units can be utilized to get reliability estimate saving time and money. This thesis focuses on the design of optimal reliability test plans for multi-component systems, and obtaining reliability estimates for systems using various techniques, such as using classical data with covariate information, ALTs, PALTs, and degradation data. Initially, the thesis addresses estimation problems in a parallel system with n independent components. The lifetime of components in the system are assumed to follow exponential distribution with parameter , where this is di erent for each component. In the literature, this is considered as constant while designing reliability test plans for parallel systems. But, in general, is not necessarily a constant; for example, the system performance may be a ected by natural covariates such as temperature, pressure, and humidity. Thus, an attempt is made to construct reliability test plans for a parallel system, where the failure rate is considered as a function of covariates. An unbiased estimator and a maximum likelihood estimator for are obtained to construct system reliability estimate. A new strategy is adopted to replace the Acceptable reliability level (ARL) and Unacceptable reliability level (URL) by Acceptable reliability interval (ARI) and Unacceptable reliability interval (URI) respectively. The advantage of this strategy is that it reduces the burden of huge rejection cost as compared to that with traditional test plans. Several examples are discussed to illustrate the resulting test plans which lead to signi cant savings in testing costs. The problem of designing reliability test plan for a series system is considered next. The focus is on designing component reliability test plans for a series system with n independent components. The advantage in this situation is that components can be tested at di erent locations, and nally, the system reliability test can be performed by incorporating individual component failure data. Based on data ob- tained from Type-II censoring, unbiased estimators for failure rates, and maximum likelihood estimator for system reliability are constructed. The design parameters are obtained by formulating an optimization problem which minimizes the maximum expected testing cost. It is observed that testing cost under Type-II censoring is random. To handle this random testing cost, an e cient algorithm is developed to minimize the total expected testing cost. Moreover, a simulation study is conducted to ensure that the derived sampling plan meets the speci ed producers and consumers risks requirements. Further, sensitivity analysis and qualitative analysis are made to study the e ect of various input parameters and to discuss the nature of reliability acceptance sampling plans. It is noted that the developed test plan has the potential of reducing testing costs of about 80% in cost reduction compared to that in existing test plans. In addition to this, it is observed that about 70% reduction in the number of components to be tested for failure, as compared to that with respect to existing plans in the literature. Bayesian statistical methods are becoming evermore popular in applied and fun- damental research. Since abundant data on failure are available in the industry in the form of prior information, the design of Bayesian reliability test plans for systems is considered in this work. Series and parallel systems with n di erent components are studied. The lifetime of each component is assumed to follow an exponential distri- bution with unknown parameter . The prior information available on is modeled by Quasi-density function, and thereby a Bayes estimator is obtained for system re- liability, based on data obtained from Type-I censoring. Examples are discussed to illustrate the resulting test plan that minimizes the total testing cost involved. It is noted that the Bayesian plan has about 70% savings in testing costs as compared to that with existing test plans in the literature. In a typical life data analysis, the reliability practitioner analyses life data (time to failure) from samples of units operating under normal working conditions in order to quantify the life characteristics of the product and make predictions about all of the units in the population. For a variety of reasons, manufacturers wish to obtain reliability results more quickly than they can when the data comes from products operating under normal conditions. An alternative for this kind of situations is to use accelerated life tests to capture life data for products under accelerated stress conditions. In this line, a novel attempt is made to construct reliability acceptance sampling plans for Weibull distribution under constant-stress PALT. The required data for constructing sampling plans are obtained from Type-II censoring. Linear and Arrhenius stress relationships are used, and MLEs of Weibull parameters and acceleration factor are obtained. Further, exact distributions of some of the pivotal quantities involved in estimating parameters in linear and Arrhenius stress relations are obtained. Since the testing cost involved is random, an expression for expected total cost is given and thereby optimal sampling plans are obtained. Some examples are also discussed to illustrate the resulting sampling plan, and the testing costs are compared as well. It is observed that plan based on Arrhenius stress model has minimum testing cost as compared to plan based on linear stress model. Also, a sensitivity analysis is carried out to analyze the e ect of a change in consumer's and producer's risks. Finally, as a substitute to the destructive testing procedure in estimating system reliability, readily available degradation data of systems are considered. An expo- nential degradation path is considered with degradation rate parameter following Weibull distribution. Unknown scale parameter of Weibull distribution is estimated. The method of Bayesian estimation is also used to estimate the parameter and thereby system reliability, by considering informative (Gamma) and non-informative (Quasi) priors for scale parameter. The standard error for estimated scale parameter cor- responding to both informative and non-informative priors are obtained using the Bootstrap method. The various approaches to system reliability estimation and test plans discussed in this thesis are suitable for realistic situations and have an advantage of savings in testing costs.