Innovative Applications of O.R.Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance
Introduction
Maintenance actions can be classified, according to their efficiency, into three types: perfect maintenance, imperfect maintenance and minimal maintenance. The assumption of perfect maintenance is only applicable to structurally simple systems, and the assumption of minimal maintenance is only applicable to highly complex systems. By contrast, it is more realistic in true experience that maintenance actions are imperfect, merely restoring a maintained system’s condition to somewhere between good-as-new and bad-as-old. To evaluate the impact of maintenance actions on a maintained system’s condition, many imperfect maintenance models have been developed. One commonly used class of imperfect maintenance models is the virtual age model introduced by Kijima (1989). The virtual age model assumes that an imperfect repair reduces a maintained system’s physical age (therein called virtual age) by an amount proportional to the physical age just before the maintenance, or by an amount proportional to the additional age accumulated since the last maintenance. Many researchers have used the concept of virtual age for modelling imperfect corrective maintenance and/or imperfect preventive maintenance; see, among others, Doyen and Gaudoin (2011), Bouguerra, Chelbi, and Rezg (2012), Dijoux and Idee (2013), Ramirez and Utne (2013), Ahmadi (2014) and Ramirez and Utne (2015). Another commonly used class of imperfect maintenance models is the improvement factor model introduced by Malik (1979). The improvement factor model assumes that an imperfect repair changes the time of the hazard rate curve to some newer time but not all the way to zero. Lin, Zuo, and Yam (2000) extended the improvement factor model, stating that an imperfect repair could change both the time and the slope of the hazard rate curve. The extended improvement factor model hence includes many documented imperfect maintenance models as special cases, e.g., the virtual age model. For recent research on the (extended) improvement factor model, the reader may refer to, e.g., Park, Chang, and Lie (2012), Xia, Xi, Zhou, and Du (2012) and Khatab, Ait-Kadi, and Rezg (2014). Other imperfect maintenance models that cannot be classified into the improvement factor model are the Brown–Proschan model (see, e.g., Doyen, 2011) and the geometric process model (see, e.g., Zhang, Xie, & Gaudoin, 2013). Reviewing works on imperfect maintenance models are given, e.g., by Wang and Pham (2006), Nakagawa (2006) and Shafiee and Chukova (2013).
We note that most of the documented work on imperfect maintenance is only concerned with age-based maintenance. When dealing with degradation-based maintenance, people typically adopt the assumption that maintenance actions are either minimal or perfect (see, e.g., Chen, Ye, Xiang, & Zhang, 2015). The issue of treating imperfect maintenance in the context of degradation-based maintenance has not received much attention and remains widely open. Zhou, Xi, and Lee (2007) employed the improvement factor model for modeling the impact of imperfect repairs on a continuously monitored deteriorating system. However, one of the major shortcomings of the improvement factor model is that it is only applicable to cases where the hazard rate function can be derived analytically. For many models, e.g. the non-stationary Wiener process, no analytical hazard rate function is available. Wang and Pham (2011) treated imperfect maintenance by lifting the degradation critical threshold proportionally. Apparently, their approach is confined to the realm of speculation. It is more desirable to develop an imperfect maintenance model by taking a physically meaningful approach. Instead of lifting the degradation critical threshold, some researchers, e.g. Nicolai, Frenk, and Dekker (2009), Van and Berenguer (2012) and Mercier and Castro (2013), treated imperfect maintenance by reducing the degradation level of a maintained system by a random amount. The uniform distribution is always employed for modeling the distribution of the random amount. We should point out that the degradation level of a maintained system before and after a repair can be exactly known (e.g., by virtue of sensors). Therefore, the amount of degradation reduced by the repair can be exactly known. Hence, the practicability of the random degradation-level-reduction paradigm is debatable. What maintenance technicians need is a practical and useful imperfect maintenance model for degradation-based maintenance. Motivated by the need of maintenance technicians and inspired by the extended improvement factor model, the broad objective of this paper is to propose an imperfect maintenance model, called random improvement factor model, for degradation-based maintenance.
The random improvement factor model is founded on the intuition that maintenance actions will change the rate of degradation of a system, other than the level of degradation. The degradation rate defined herein is the rate of accumulation of degradation (see, e.g., Meeker & Escobar, 1998). Repairs slowing down degradation rate can be easily visualized in the context of, e.g., coating operations. In order to slow down the deteriorating process, steel structures are usually protected by certain organic coating systems (Perrin, Merlatti, Aragon, & Margaillan, 2009). The degradation level after each coating operation may or may not be changed. Another type of maintenance is lubricating rotating gears, by which the wearing processes of the gears will slow down. The degradation level after lubrication remains unchanged. One distinguishing feature of the random improvement factor model is the introduction of a latent variable, taking into account the fact that each maintenance action should have a different degree of impact on the rate of degradation. Accordingly, the filtering technique is utilized for dynamically estimating each realization of the latent variable. For illustrative purpose, the Wiener process is employed due to its adaptability in modeling degradation phenomena. The Wiener process (in its many forms) has been applied in Meeker and Escobar (1998) to model the size of fatigue crack as a function of the number of cycles, in Wang (2010) to model the degradation of bridge beams due to chloride ion ingression, and in Ye, Wang, Tsui, and Pecht (2013) to model the wear of magnetic heads used in hard disk drives and the light output of a light emitting diode. Applications of the Wiener process are also reported in Nikulin, Limnios, Balakrishnan, Kahle, and Huber-Carol (2010), Bian and Gebraeel (2012), Son, Fouladirad, Barros, Levrat, and Iung (2013), Si, Chen, Wang, Hu, and Zhou (2013) and Si, Wang, Hu, and Zhou (2014), to name a few. We shall underscore that, after appropriate modifications, the following procedure can be readily applied to other stochastic processes.
The remainder of the paper is organized as follows. Section 2 is devoted to presenting the random improvement factor model (in Section 2.1) and a generic maintenance scheme (in Section 2.2). Section 3 is devoted to real-time updating the degradation rate function of the maintained system (in Section 3.1) and the estimates of other fixed model parameters (in Section 3.2). Section 4 gives numerical examples to show the applicability and competence of the advanced techniques. Section 5 outlines concluding remarks and future work.
Section snippets
Random improvement factor model
Let v(t) be a differentiable, non-negative, real-valued function of t( ≥ 0) with v(0) = 0. A Wiener process, denoted by {Xt, t ≥ 0}, with drift function v(t) and variance coefficient σ( > 0) is defined by Xt = v(t) + σWt; see, e.g., Si, Wang, Hu, Zhou, and Pecht (2012), Bian and Gebraeel (2012) and Son et al. (2013). {Wt, t ≥ 0} is the standard Brownian motion. The Wiener process has independent and normally distributed random increments. That is, for all 0 ≤ s < t, Xt − Xs is independent of Xs
Parameter estimation
For the practical implementation of the random improvement factor model, the fundamental problem is how to estimate unknown parameters. We note that the degradation-rate-reduction factor is a latent random variable. By contrary, the other unknown model parameters (e.g., σ) are fixed. Therefore, different methods are developed for evaluating different types of model parameters.
Numerical examples
The competence and robustness of the Kalman filter and QMC method are evidenced via simulated data, and the utility of the random improvement factor model is revealed via a real data set. Simulated degradation data are generated via the following method, called DG method. At time ▵, we randomly simulate a degradation datum, denoted by x1, from the normal distribution N(λ▵θ, σ2▵).
- •
If an imperfect repair is then performed at time ▵, the value of the scale parameter changes, immediately after the
Conclusions
In this paper, we developed an imperfect maintenance model for systems whose sensor information can be modeled by stochastic processes. There are three main contributions of this paper. The first contribution is the modeling of imperfect maintenance via the degradation rate function instead of the hazard rate function. Our model is more suitable and practical for degradation-based maintenance. The second contribution is the introduction of a random improvement factor. The assumption of random
Acknowledgments
We are grateful to the editor and three reviewers for their critical and constructive comments that have considerably helped in the revision of an earlier version of the paper.
References (50)
- et al.
A decision model for adopting an extended warranty under different maintenance policies
International Journal of Production Economics
(2012) - et al.
Condition-based maintenance using the inverse gaussian degradation model
European Journal of Operational Research
(2015) - et al.
Design of inferential sensors in the process industry: A review of bayesian methods
Journal of Process Control
(2013) - et al.
Quasi-monte carlo integration
Journal of Computational Physics
(1995) - et al.
Modelling and optimizing imperfect maintenance of coatings on steel structures
Structural Safety
(2009) - et al.
Stress-reducing preventive maintenance model for a unit under stressful environment
Reliability Engineering & System Safety
(2012) - et al.
Degradation study of polymer coating: Improvement in coating weatherability testing and coating failure prediction
Progress in Organic Coatings
(2009) - et al.
Use of dynamic bayesian networks for life extension assessment of ageing systems
Reliability Engineering & System Safety
(2015) Error trends in quasi-monte carlo integration
Computer Physics Communications
(2004)- et al.
Maintenance models in warranty: A literature review
European Journal of Operational Research
(2013)
Specifying measurement errors for required lifetime estimation performance
European Journal of Operational Research
When are quasi-monte carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Wiener processes with random effects for degradation data
Journal of Multivariate Analysis
Modeling and optimizing maintenance schedule for energy systems subject to degradation
Computers & Industrial Engineering
Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation
Reliability Engineering & System Safety
Optimal maintenance scheduling for a complex manufacturing system subject to deterioration
Annals of Operations Research
Computing and updating the first-passage time distribution for randomly evolving degradation signals
IIE Transactions
An overview of existing methods and recent advances in sequential Monte Carlo
Proceedings of the IEEE
The theory of stochastic processes
Classes of virtual age models adapted to systems with a burn-in period
IEEE Transactions on Reliability
On the brownproschan model when repair effects are unknown
Applied Stochastic Models in Business and Industry
Modeling and assessment of aging and efficiency of corrective and planned preventive maintenance
IEEE Transactions on Reliability
Residual-life distributions from component degradation signals: A Bayesian approach
IIE Transactions
Random number generation and monte carlo methods
Availability optimisation for stochastic degrading systems under imperfect preventive maintenance
International Journal of Production Research
Cited by (131)
Risk evolution of crude oil pipeline under periodic maintenance based on dynamic bayesian network
2024, Journal of Loss Prevention in the Process IndustriesThe effect of model misspecification of the bounded transformed gamma process on maintenance optimization
2024, Reliability Engineering and System SafetyOptimal condition-based maintenance policy for multi-component repairable systems with economic dependence in a finite-horizon
2024, Reliability Engineering and System SafetyOptimal maintenance policy for systems with environment-modulated degradation and random shocks considering imperfect maintenance
2023, Reliability Engineering and System SafetyOpportunistic maintenance strategy optimization considering imperfect maintenance under hybrid unit-level maintenance strategy
2023, Computers and Industrial EngineeringMaintenance optimization of a system subject to two-stage degradation, hard failure, and imperfect repair
2023, Reliability Engineering and System Safety