Optimal condition-based maintenance policy with delay for systems subject to competing failures under continuous monitoring
Introduction
With the rapid development of technologies to monitor and measure system deterioration, condition-based maintenance (CBM) is playing a more important role to maintain systems that suffer from measurable degradations (Lu and Meeker, 1993). To facilitate CBM, maintenance operations are scheduled based on the online measurements of quality characteristics, such as degradation levels, instead of following a fixed maintenance schedule or age-based policy (Grall et al., 2002, Huynh et al., 2018). For modern manufacturing systems, the planning of maintenance based on online status is of great significance to ensure production efficiency and products’ quality.
In most existing studies of CBM, an implicit assumption is that systems only suffer from soft failures caused by degradation, i.e., the system fails if and only if the degradation level exceeds a certain threshold. For an overview of CBM under stochastic degradation models, one can be referred to Alaswad and Xiang (2017). In recent literature, competing failure models have been intensely studied for CBM (Cha, Sanguesa, & Castro, 2016). In Rafiee et al., 2015, Castro et al., 2015, Caballé et al., 2015, the mixed failure model was extended to multiple dependent degradations accompanied by random shocks. Zhu, Fouladirad, and Bérenguer (2015) introduced covariates into the mixed failure model and optimized the CBM policy. Zhao, Gaudoin, Gaudoin, Doyen, and Xie (2018) considered both positive and negative influences of inspections to optimize CBM policies. For multi-component systems, maintenance values and component importance measures were employed used to optimize the CBM policies (Liu, Xu, Xie, & Kuo, 2014). In Peng and Van Houtum (2016), CBM and production policies are jointly optimized to minimize the average long-run cost. Other CBM research topics include using new classes of stochastic processes to model degradation (Chen, Ye, Xiang, & Zhang, 2015) and failure processes (Lee and Cha, 2016), CBM policies for systems with multiple dependent components (Liu, Xie, & Kuo, 2016), etc.
Most studies on CBM assume that the degradation levels are measured periodically, and decision variables are the inspection interval and preventive maintenance threshold. As the monitoring cost decreases, many systems (e.g., automobiles and advanced processing machine tools) are now equipped with monitoring mechanisms to provide continuous quality information. Continuous monitoring can also be realized by discrete measurements with negligible inspection intervals. However, the research of CBM for systems under continuous monitoring are relatively scant in the literature. Marseguerra, Zio, and Podofillini (2002) used Monte Carlo simulation to find the optimal CBM policy for single-unit systems under continuous monitoring. Liao, Elsayed, and Chan (2006) utilized imperfect repair models for continuously monitored degrading systems and determined the optimal maintenance threshold. Note that only degradation failures were considered in these studies. Tian, Jin, Wu, and Ding (2011) proposed a CBM optimization method for wind power systems under continuous monitoring. Liu et al. (2013) considered multiple failure modes for degrading systems under continuous monitoring and obtained the optimal alarm threshold to minimize the maintenance cost or maximize the system availability.
In CBM problems for continuously monitored system, the general objective is to choose the optimal maintenance threshold to minimize the maintenance cost. The threshold is an alarm threshold that reminds the field operators or reliability engineers to maintain the system. In real engineering practice, it is likely that a delay occurs in maintenance operations for some practical reasons. One of such cases is that at the instant when the alarm is triggered, the system may be in a state where it cannot be stopped working immediately. For example, the alarm for problems in a manufacturing system is usually triggered during the production. It can be very difficult and costly to stop the machine immediately, and the maintenance operation usually delays for a moment. Furthermore, to maintain complicated machines typically need external professions, which is subject to nonnegligible lead time for preparation. Existing research that considered delay time in maintenance operations for degrading systems is very limited. In Bérenguer, Grall, Dieulle, and Roussignol (2003), the maintenance was delayed by a fixed lead time and the alarm threshold was selected to minimize the asymptotic unavailability. Meier-Hirmer, Riboulet, Sourget, and Roussignol (2009) proposed to minimize the long-run maintenance cost by considering a fixed intervention delay with application to track maintenance. However, the fact that delay time in maintenance operations could be random is rarely taken into account. In Yang, Ma, Zhai, and Zhao (2016), system replacement is assumed to be postponed if failures occur too early in an inspection cycle and inspection/maintenance policies are investigated for single-component systems. He, Maillart, and Prokopyev (2017) studied the optimal planning of preventive maintenance with unpunctual maintenance actions.
Different from previous works, the following aspects are considered simultaneously in the paper: (1) the system is continuously monitored; (2) failures result from two sources, i.e., degradation and random fatal shocks, and the intensity of shocks depends on the degradation level; (3) there is a random delay time in maintenance operations after the alarm is triggered. Based on these assumptions, optimal CBM policy is selected to minimize the long run cost rate.
The remainder of the paper is organized as follows. Section 2 introduces the basic assumptions and probabilistic modeling of the competing failures. Section 3 describes the condition-based maintenance policy and derives the expected cost rate function analytically, which is used as the objective function. In Section 4, an extension that addresses the variability of maintenance cost due to parameter uncertainty is presented. Numerical examples are presented with sensitivity analysis in Section 5. Finally, we give conclusions and future perspectives in Section 6.
Section snippets
Modeling of competing failures
A single-unit system with dependent degradation and external shocks is studied in this paper. When the degradation level exceeds the failure threshold , a “degradation failure” occurs. Additionally, when external shock occurs prior to the degradation failure, the system fails immediately, i.e., a “shock failure” occurs.
Costs description and objective function
To maintain a degrading system under continuous monitoring, there is no need to conduct periodic inspections. The preventive maintenance can be planned based on the online system state. Moreover, we assume that the system is nonrepairable, i.e., replacement is the only approach to system maintenance. Since there is a random delay time following the alarm, it is likely that the system fails before it is preventively replaced. If any failure occurs, the system is correctively replaced with a new
Extension: maintenance cost considering parameter uncertainty
In practice, maintenance policies usually need to be determined before the systems are put into the market and operated in field conditions. For the problem described in this paper, the alarm threshold also needs to be specified and programmed into the system prior to its usage. Therefore, it is necessary to study the robustness of maintenance policy under parameter uncertainty (Qiu, Cui, Shen, & Yang, 2017). Manufacturers may not be very confident with the pilot system parameters to
Numerical example
To illustrate the proposed approach in this paper, without loss of generality, we analyze a numerical example with the following assumptions:
- (1)
The degradation process is well modeled by a gamma process with parameters .
- (2)
The fail threshold ; the failure rate change threshold .
- (3)
Failure rates .
- (4)
The distribution of delay time is a uniform distribution between 0 and 12, i.e., .
- (5)
The cost settings for events: .
Recall
Conclusions
In this paper, a maintenance problem for continuously monitored system is studied by introducing random delay time to the maintenance operations. A condition-based maintenance policy is developed by minimizing the expected cost rate. The analytical long-term cost rate is derived under dependent shock and degradation risks. Afterward, numerical examples along with sensitivity analyses provide illustration and quantitative comparisons on optimal alarm thresholds and objective cost rates under
Funding
This work was supported in part by the Research Grants Council of Hong Kong under a theme-based project Grant (T32-101/15-R) and a General Research Fund (CityU 11203815) and in part by the National Natural Science Foundation of China under a Key Project Grant 71532008.
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