maintenance factor for corrective maintenance (ar) and the imperfect ... system (RS) is a system which, after failure, can be restored to a functioning condition.
Parameter Estimation for a Repairable System under Imperfect Maintenance Pingjian Yu, University of Arkansas Joon Jin Song, Ph.D., University of Arkansas C. Richard Cassady, Ph.D., University of Arkansas Keywords: Bayesian Analysis, Imperfect Maintenance, Parameter Estimation, Reliability and Maintainability SUMMARY & CONCLUSIONS Estimation of reliability and maintainability parameters is essential in modeling repairable systems and determining maintenance policies. However, because of the aging of repairable systems under imperfect maintenance, failure times are neither identically nor independently distributed, which makes parameter estimation difficult. In this paper, we apply Bayesian methods for estimation of reliability and maintainability parameters based on historical reliability and maintainability (RAM) data. We assume the first failure of the repairable system follows a Weibull probability distribution. The repairable system experiences Kijima Type I imperfect corrective maintenance and Kijima Type I imperfect preventive maintenance. Using a Bayesian perspective, we estimate four parameters for this repairable system: the shape parameter of the Weibull probability distribution (β), the scale parameter of the Weibull distribution (η), the imperfect maintenance factor for corrective maintenance (ar) and the imperfect maintenance factor for preventive maintenance (ap). The proposed method is illustrated with simulated RAM data. 1 INTRODUCTION Professionals in all industries and the military are faced with challenges associated with performing maintenance actions on and optimizing maintenance planning for their repairable systems. A repairable system (RS) is a system which, after failure, can be restored to a functioning condition by some maintenance action other than replacement of the entire system. Note that replacing the entire system may be an option, but it is not the only option. For a repairable system, time to failure depends on both the life distribution (the probability distribution of the time to first failure) and the impact of corresponding maintenance actions performed on the system. Whereas there are some physical models to describe time to failure for certain repairable systems, statistical models are convenient for decision makers to model repairable systems based on historical reliability and maintainability (RAM) data without knowing the failure mechanism (such as wear, fatigue, temperature, etc.). The literature on the use of statistical models for
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evaluating the performance of RS and optimizing maintenance plans for RS is extensive. See McCall (1965)[1], Pierskalla and Voelker (1976)[2], Valdez-Florez and Feldman (1989)[3], Cho and Parlar (1991)[4], Dekker (1996)[5], and Wang (2002)[6] for surveys of this literature. Generally, statistical models are constructed in terms of reliability and maintainability parameters. Thus, accurate estimation of reliability and maintainability parameters is essential in using these models to optimize maintenance strategies. The Weibull probability distribution is the most widely used life distribution model. The Weibull probability distribution has two parameters, the shape parameter β and the scale parameter η. Since β and η represent the reliability characteristics of a system (either repairable or nonrepairable), we take these two parameters as the reliability parameters of the repairable systems of interest. Maintenance actions performed on a RS can be categorized into two groups: corrective maintenance (CM) actions and preventive maintenance (PM) actions. CM actions are performed in response to system failures, and they could correspond to either repair or replacement activities. PM actions are not performed in response to RS failure, but they are intended to delay or prevent system failures. After a maintenance action (CM or PM), the RS begins a new period of proper function. The duration of this period depends on the time to first failure distribution and the assumed impact of maintenance on the RS. According to Pham and Wang (1996)[7], the impact of maintenance can be classified into one of five categories: (1) Perfect: The maintenance action restores the RS to an “as good as new” condition. (2) Minimal: The maintenance action restores the RS to an “as bad as old” condition, i.e., the RS is properly functioning but its effective age is the same as at the instant maintenance was initiated. (3) Imperfect: The maintenance action restores the RS to a condition between “as good as new” and “as bad as old”. (4) Worse: The maintenance action makes the RS function properly, but its effective age is greater than at the instant maintenance was initiated. (5) Worst: The maintenance action makes the RS fail.
The focus in this paper is on imperfect maintenance, but note that perfect maintenance and minimal maintenance are the two extreme cases of imperfect maintenance. To analyze the behavior of a RS under imperfect maintenance, a mathematical model is needed to describe the impact of imperfect maintenance. A wide variety of imperfect maintenance models have been studied in the literature. See Pham and Wang (1996) for a survey of these models. Of these models, the focus in this paper is on the Kijima Type I virtual age model. Initially proposed by Kijima (1988)[8], the Kijima Type I virtual age model indicates that V n = V n −1 + aX n where Xn denotes the length of the nth period of RS function, n = 1, 2, … , Vn denotes the virtual (effective) age of the RS after the nth maintenance action (V0 = 0), n = 0, 1, …, and a denotes the imperfect maintenance factor (0 ≤ a ≤ 1). Under the Kijima Type I virtual age model, each maintenance action removes a portion (1−a) of the age accumulated by the RS during its last period of function. Note that a = 0 is equivalent to minimal maintenance and a = 1 is equivalent to perfect maintenance. In this paper we consider a RS that experiences Kijima Type I imperfect CM and Kijima Type I imperfect PM. The imperfect maintenance factors for CM and PM are ar and ap respectively. Since ar and ap represent the maintainability characteristics of a repairable system, we take these two parameters as the maintainability parameters of the repairable systems of interest. The objective of this paper is to estimate the reliability parameters (β and η) and the maintainability parameters (ar and ap) simultaneously from historical RAM data using a Bayesian perspective. Unlike classical statistical inference methods such as rank or least squares methods, method of moments, and maximum likelihood estimation (MLE), which assume that the parameters to be estimated are fixed values, Bayesian methods assume that the parameters to be estimated can be described using a probability distribution. The literature on estimating reliability parameters (β and η) for non-repairable systems is abundant from both classical and Bayesian perspectives. Like in non-repairable systems, the first failure of identical repairable systems is independent of ar and ap. So we can use those methods in the literature to estimate the reliability parameters (β and η) from the data of first failures of identical repairable systems. However, in practice, it is very difficult to collect large amounts of such data to make this estimation precisely. On the other hand, large RAM data after maintenance (these failure data are dependent on ar and ap) is easy to obtain. Some recent studies have been done to estimate reliability and maintainability parameters from RAM data. From a classical perspective, Crow (1990)[9] estimates reliability parameters (β and η) under minimal repair (ar = 1). Jack (1998)[10] estimates reliability and maintainability parameters (β, η, ar and ap) under Kijima Type I imperfect CM and Kijima Type I imperfect PM (the model allows PM to remove age accumulated since the last PM action instead of the last
functioning period). Seo (2003)[11] estimates reliability parameters (β and η) under minimal repair (ar = 1) and perfect PM (ap = 0). Gasmi (2003)[12] estimates reliability and maintainability parameters (β, η and ar) under two types of CM, one is minimal CM and the other is Kijima Type I imperfect CM. Mettas and Zhao (2005)[13] estimate reliability and maintainability parameters (β, η and ar) under Kijima Type II imperfect CM. From a Bayesian perspective, Chen (2000)[14] estimates reliability parameters (β and η) under minimal repair (ar = 1) and perfect PM (ap = 0). Dayanik (2002)[15] estimates reliability parameters (β and η) under minimal repair (ar = 1). Suppose historical RAM data on the RS is available. This historical data includes data on one repairable system for a period of time that includes at least one imperfect CM action and at least one imperfect PM action. In this paper, we develop a Bayesian method for using this data to estimate the RAM parameters {β, η, ar, ap} of the RS. The remainder of this paper is organized as follows: In section 2, we define the system of interest and the data collection schema. In section 3, we model the problem defined in section 2 using a Bayesian method and apply a sampling algorithm to solve the model. In section 4, we develop a numerical example of parameter estimation from simulated RAM data. In section 5, we summarize our research and give some directions of future studies. 2 SYSTEM DEFINITION AND DATA COLLECTION SCHEMA Consider a RS that experiences Kijima Type I imperfect CM and Kijima Type I imperfect PM. The PM policy is that during the ith period of RS function, if the RS survives τi time units, a PM action is performed; otherwise a CM action is performed. The time to first failure of this repairable system (Y1) follows a Weibull probability distribution with shape parameter β > 0 and scale parameter η > 0. Let F1(y) denote the cumulative distribution function (CDF) of Y1. Then, ⎡ ⎛ y ⎞β ⎤ F1( y ) = 1 − exp ⎢− ⎜⎜ ⎟⎟ ⎥ ⎢ ⎝η ⎠ ⎥ ⎣ ⎦ If Y10, 0
