Olivier Gaudoin

Classes of imperfect repair models based on reduction of failure intensity or virtual age

Abstract This article proposes two new classes of imperfect repair models. The (conditional) failure intensity before the first repair is a continuous function of time. The repair effect is characterized by the change induced on the failure intensity before and after failure. In the first class of models, repair effect is expressed by a reduction of failure intensity. In the second class, repair effect is expressed by a reduction of the system virtual age. In each case, several particular cases are studied, which take into account the possibility of a markovian memory property. For almost every model studied, there exists a minimal wear intensity, i.e. a maximal lower bound for failure intensity. The classification presented here involves existing models and allows the proposition of new ones. The models are compared in terms of wear-out speed. Finally, a numerical statistical study on the quality of the model parameters estimators is presented.

A simple goodness-of-fit test for the power-law process, based on the Duane plot

The PLP (power-law process) or the Duane model is a simple model that can be used for both reliability growth and reliability deterioration. GOF (goodness-of-fit) tests for the PLP have attracted much attention. However, the practical use of the PLP model is its graphical analysis or the Duane plot, which is a log-log plot of the cumulative number of failures versus time. This has been commonly used for model validation and parameter estimation. When a plot is made, and the coefficient of determination, R/sup 2/, of the regression line is computed, the model can be tested based on this value. This paper introduces a statistical test, based on this simple procedure. The distribution of R/sup 2/ under the PLP hypothesis is shown not to depend on the true model parameters. Hence, it is possible to build a statistical GOF test for the PLP. The critical values of the test depend only on the sample size. Simulations show that this test is reasonably powerful compared with the usual PLP GOF tests. It is sometimes more powerful, especially for deteriorating systems. Implementing this test needs only the computation of a coefficient of determination. It is much easier than, for example, computing an Anderson-Darling statistic. Further study is needed to compare more precisely this new test with the existing ones. But the R/sup 2/ test provides a very simple and useful objective approach for decision making with regard to model validation.

Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance

The notion of imperfect maintenance has spawned a large body of literature, and many imperfect maintenance models have been developed. However, there is very little work on developing suitable imperfect maintenance models for systems outfitted with sensors. Motivated by the practical need of such imperfect maintenance models, the broad objective of this paper is to propose an imperfect maintenance model that is applicable to systems whose sensor information can be modeled by stochastic processes. The proposed imperfect maintenance model is founded on the intuition that maintenance actions will change the rate of deterioration of a system, and that each maintenance action should have a different degree of impact on the rate of deterioration. The corresponding parameter-estimation problem can be divided into two parts: the estimation of fixed model parameters and the estimation of the impact of each maintenance action on the rate of deterioration. The quasi-Monte Carlo method is utilized for estimating fixed model parameters, and the filtering technique is utilized for dynamically estimating the impact from each maintenance action. The competence and robustness of the developed methods are evidenced via simulated data, and the utility of the proposed imperfect maintenance model is revealed via a real data set.

Modeling and Assessment of Aging and Efficiency of Corrective and Planned Preventive Maintenance

The paper presents a general framework for the simultaneous modeling and assessment of aging and maintenance efficiency, for complex repairable systems. Both corrective and planned preventive maintenance actions are considered. The proposed framework takes into account both imperfect maintenance efficiency, through for instance virtual age models, and the usual preventive maintenance policies, such as periodic, sequential, age-dependent, and failure limit. The main properties of the general model are studied, and reliability indicators are derived. The model parameters are estimated by the maximum likelihood method. Finally, we present an application to real data sets issued from electricity production systems.

Confidence intervals for the scale parameter of the power-law process

The power-law process (PLP) is a two-parameter model widely used for modeling repairable system reliability. Results on exact point estimation for both parameters as well as exact interval estimation for the shape parameter are well known. In this paper, we investigate the interval estimation for the scale parameter. Asymptotic confidence intervals are derived using Fisher information matrix and theoretical results by Cocozza-Thivent (1997). The accuracy of the interval estimation for finite samples is studied by simulation methods.

More on the Mis-Specification of the Shape Parameter with Weibull-to-Exponential Transformation

When lifetimes follow Weibull distribution with known shape parameter, a simple power transformation could be used to transform the data to the case of exponential distribution, which is much easier to analyze. Usually, the shape parameter cannot be known exactly and it is important to investigate the effect of mis-specification of this parameter. In a recent article, it was suggested that the Weibull-to-exponential transformation approach should not be used as the confidence interval for the scale parameter has very poor statistical property. However, it would be of interest to study the use of Weibull-to-exponential transformation when the mean time to failure or reliability is to be estimated, which is a more common question. In this paper, the effect of mis-specification of Weibull shape parameters on these quantities is investigated. For reliability-related quantities such as mean time to failure, percentile lifetime and mission reliability, the Weibull-to-exponential transformation approach is generally acceptable. For the cases when the data are highly censored or when small tail probability is concerned, further studies are needed, but these are known to be difficult statistical problems for which there are no standard solutions. Copyright

An example of integrated approach to technical and economic optimization of maintenance

Abstract This paper presents a case study of technical and economic optimization of the periodicity of predetermined preventive maintenance actions carried out on a repairable industrial system from an EDF electric power plant. This analysis is conducted with the MARS software tool (MARS for “maintenance assessment of repairable systems”), developed jointly by Grenoble University and EDF R&D. The innovative aspect of this work lies in the integrated approach that is used, combining two steps. A first estimation step retrospectively assesses maintenance effect on system reliability. A second simulation step predicts the behavior of the maintained system over the time period set as an objective by the operator. The different stages of the case study are described in detail with elaborated considerations about optimization of the periodicity of preventive maintenance.

A Bivariate Maintenance Policy for Multi-State Repairable Systems With Monotone Process

This paper proposes a sequential failure limit maintenance policy for a repairable system. The objective system is assumed to have k+1 states, including one working state and k failure states, and the multiple failure states are classified potentially by features such as failure severity or failure cause. The system deteriorates over time and will be replaced upon the Nth failure. Corrective maintenance is performed immediately upon each of the first (N-1) failures. To avoid the costly failure, preventive maintenance actions will be performed as soon as the systems reliability drops to a critical threshold R. Both preventive maintenance and corrective maintenance are assumed to be imperfect. Increasing and decreasing geometric processes are introduced to characterize the efficiency of preventive maintenance and corrective maintenance. The objective is to derive an optimal maintenance policy (R*,N*) such that the long-run expected cost per unit time is minimized. The analytical expression of the cost rate function is derived, and the corresponding optimal maintenance policy can be determined numerically. A numerical example is given to illustrate the theoretical results and the maintaining procedure. The decision model shows its adaptability to different possible characteristics of the maintained system.

U-plot method for testing the goodness-of-fit of the power-law process

This paper shows that the u plot method can be used as a statistical procedure for testing the fit of the Power-Law-Process (PLP), also called the Duane model. The u-plot was initially presented as a graphical tool for validating software reliability models. Mathematical derivations are given here to justify the use of u-plot as a statistical goodness-of-fit test for the PLP. These results give theoretical explanations to Downs and Scott sim¬ulation results. Methods and tools are suggested to further investigations which should lead to proving that the u-plot method can be considered as a common statistical goodness-of-fit test for several reliability growth mod¬els. The u-plot test is compared to classical PLP goodness-of-fit tests for different alternative models, and appears to perform very well.

Bayesian Analysis of ARA Imperfect Repair Models

This article proposes a Bayesian analysis of a class of imperfect repair models, the ARA models. The choice of prior distributions and the computation of posterior distributions are discussed. The presentation is unified for all ARA models and many kinds of possible priors. A numerical study on the quality of the Bayesian estimators is presented, as well as a comparison with the maximum likelihood estimators. Finally, the approach is applied to a real data set.