Reliability Testing, Maintenance and Warranty Policies Considering Environmental Covariates


Student thesis: Doctoral Thesis

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Awarding Institution
Award date30 Jul 2018


Reliability characteristics of many industrial systems are remarkably influenced by their working conditions, which can be modeled via environmental covariates. These covariates provide the possibility to test certain systems with reliability experiments and further provide useful information to support reliability-related managerial decisions. For both lifetime and degradation models, the influence of environmental covariates has been intensely investigated in the literature to address various reliability-related topics, such as reliability assessment, prognostics, maintenance and warranty management. With the development in modern sensor and monitoring technologies, environmental covariates can be precisely measured and fully utilized to overcome typical issues in reliability studies. The thesis, consisting of six studies, investigate the reliability testing, maintenance and warranty issues for highly reliable products with environmental covariates.

First, we study two problems concerning accelerated reliability tests that employ stress covariates to predict reliability characteristics:

Robust accelerated life tests under model uncertainty. Accelerated life testing (ALT) is commonly used to predict the lifetime of a product at its use stress by subjecting test units to elevated stress conditions that accelerate the occurrence of failures. For new products, the selection of an acceleration model for planning optimal ALT plans is challenging due to the absence of historical lifetime data. The misspecification of an ALT model can lead to considerable errors when it is used to predict the product’s life quantiles. To deal with this, we propose a two-stage Bayesian approach to constructing ALT plans and predicting lifetime quantiles. At the first stage, the ALT plan is optimized based on the prior information of candidate models under a modified V-optimality criterion that incorporates both asymptotic prediction variance and squared bias. A Bayesian model averaging (BMA) framework is used to derive the posterior model and the posterior distribution for the life quantile of interest under use stress. If the obtained test data cannot provide satisfactory model selection results, an adaptive second-stage test is conducted based on the posterior information from the first stage. Revisited numerical examples demonstrate the efficiency and robustness of the resulting Bayesian ALT plans by comparing with the plans derived from previous methods.

Accelerated degradation tests under competing failure models. Accelerated degradation tests (ADT) have been widely used to assess the reliability of products with a long lifetime. For many products, environmental stress not only accelerates their degradation rate but also elevates the probability of traumatic shocks. When random traumatic shocks occur during an ADT, it is possible that the degradation measurements cannot be taken afterward, which brings challenges to reliability assessment. Here, we propose an ADT optimization approach for products suffering from both degradation failures and random shock failures. The degradation path is modeled by a Wiener process. Under various stress levels, the arrival process of random shocks is assumed to be a non-homogeneous Poisson process. Parameters of acceleration models for both failure modes need to be estimated from the ADT. Three common optimality criteria based on the Fisher information are considered and compared to optimize the ADT plan under a given number of test units and a pre-determined test duration. Optimal two- and three-level optimal ADT plans are obtained by numerical methods. We use the general equivalence theorems to verify the global optimality of ADT plans. A numerical example is presented to illustrate the proposed methods. The result shows that the optimal ADT plans in the presence of random shocks differ significantly from the traditional ADT plans. Sensitivity analysis is carried out to study the robustness of optimal ADT plans with respect to the changes in planning input.

The idea of incorporating covariates into reliability tests can be adopted to model multi-component systems. The following work explores the reliability analysis for parallel load-sharing systems with degrading components:

Degradation modeling for load-sharing systems. Load-sharing systems are commonly applied in various industries. In such systems, component failures result in an elevated workload of surviving components, which typically accelerates the failure of the whole system. For load-sharing systems with identical components subject to continuous degradation, a reliability modeling and analysis framework are proposed. It is assumed that the components in the system suffer from degradation through an additive impact under increased workload caused by consecutive failures. A log-linear link function is used to describe the relationship between the degradation rate and load stress levels. By assuming the component degradation is well modeled by a step-wise drifted Wiener process, we construct maximum likelihood estimators (MLE) for unknown parameters and related reliability characteristics by combining analytical and numerical methods. Approximate initial guesses are proposed to lessen the computational burden in numerical estimation. The asymptotic distribution of MLE is given in the form of multivariate normal distribution with the aid of Fisher information. Alternative confidence intervals are provided by bootstrapping methods. A simulation study with various sample sizes and inspection intervals is presented to analyze the estimation accuracy. Finally, the proposed approach is illustrated by track degradation data of railway tracks.

Afterward, based on degradation test data, optimal condition-based maintenance policy and warranty policy are explored for continuous degrading systems:

Maintenance decisions based on reliability test data. Maintenance can be classified into two types according to its scheduling methods: time-based maintenance (TBM) and condition-based maintenance (CBM). As technologies for monitoring and measuring deterioration advances, CBM is playing a more important role to maintain highly reliable systems that suffer measurable degradation. In CBM, maintenance operations are scheduled based on the online measure of quality characteristics, such as degradation level, instead of a fixed schedule or age policy. As part of research works in the thesis, a novel maintenance model is proposed for single-unit systems with atypical degradation path of which the pattern is changed by inspections. After each inspection, the system degradation is assumed to decrease by a random value. Meanwhile, the degrading rate is elevated due to the inspection. Considering the double effects of inspections, we develop a parameter estimation procedure for such systems from experimental data obtained via degradation tests with environmental covariates. Next, the inspection and replacement policy is optimized with the objective to minimize the expected long-run cost rate (ELRCR). Inspections are assumed to be non-periodically scheduled. A numerical algorithm that combines analytical and simulation methods is presented to evaluate the ELRCR. Afterward, we investigate the robustness of maintenance policies for such systems by taking the parameter uncertainty into account with the aid of large-sample approximation and parametric bootstrapping. The application of the proposed method is illustrated using the motivating example from electric industry.

Warranty cost optimization under dynamic environments. For new products that have not been put on the market, manufacturers usually want to predict the warranty cost to forecast its influence on future profit. In the test phase of new products, accelerated life tests (ALT) are commonly used to predict the lifetime under use condition. By analyzing ALT experimental data, we present a framework to predict the warranty cost and risk under one-dimensional warranty. Two sources of variability are considered to make inferences of predicted warranty cost: the uncertainty of estimated parameters from ALT data and the variability of field conditions. Under these assumptions, the expected warranty cost and warranty risk are computed by Markov chain Monte Carlo (MCMC) sampling based on the approximated lifetime distribution. We assume that the warranty guarantees imperfect repairs. The framework could be easily repeated for ALT data based on log-location-scale lifetime distributions and both constant-stress and step-stress ALT data. Compared with original Monte Carlo (MC) simulation, the proposed method provides comparable prediction accuracy with significantly less computational effort. A numerical example with sensitivity analysis is given to illustrate the effectiveness of the proposed methods.

Further, we propose a warranty cost optimization framework based on degradation data with environmental covariates within a finite warranty period under the assumption of imperfect repairs. The number of warranty claims is given in the analytical form. We also consider the two sources of variability as in the last work. Additionally, the imperfect repairs are assumed to be random. The warranty cost for a single repair is assumed to be associated with the improvement factor of imperfect repairs, which is modeled by a truncated normal distribution. Optimal imperfect repair policy is obtained by minimizing the expected warranty cost for each sold product. Numerical examples show that the proposed method to evaluate warranty claims significantly outperforms simulation methods. A real application example is also presented to illustrate the proposed framework.

Through a series of works related to the reliability issues of highly reliable products with environmental covariates, the thesis aims to bridge several gaps that are of interest to researchers and practitioners. The findings via the research proposed will provide scientific insights to support managerial decision making at multiple stages in reliability management.