An imperfect age-based and condition-based opportunistic maintenance model for a two-unit series system

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Highlights

  • An imperfect opportunistic maintenance policy for a two-unit system is proposed.

  • Deterioration levels and usage of units are utilized to repair the whole system.

  • Semi-Markov decision process is used to solve the optimal optimization problem.

  • Maintenance cost is minimized by optimizing the maintenance thresholds of units.

Abstract

Maintenance strategies have been studied extensively and applied successfully to reduce the operation costs of industrial assets. However, the study for heterogeneous two-unit systems is rarely based on the usage and condition information of units, simultaneously. This paper originally proposes a novel imperfect opportunistic maintenance model for a two-unit series system considering random repair time and two types of failures, where unit 1 and unit 2 are respectively subject to soft failure and hard failure. The system maintenance actions are performed not only based on the deterioration level of unit 1 but also on the usage of unit 2. To utilize the economic dependence between the two units, the opportunistic maintenance policy is adopted in the series system. When a unit fails or is being repaired, the other unit has an opportunity to be simultaneously maintained once it meets the requirement of opportunistic maintenance. Herein, opportunistic maintenance action is imperfect maintenance, which means that the deterioration level of unit 1 and the usage of unit 2 restore to a lower level than the as-good-as new status after repair. Nonetheless, once both units meet the requirement of replacement actions, they will be perfectly replaced by new ones. The objective is to minimize the system average maintenance cost by optimizing the thresholds of opportunistic maintenance and perfect replacement of two units. Finally, an illustrative example is presented to demonstrate the superiority of the proposed maintenance model.

Introduction

Maintenance strategies are of great importance to reduce maintenance costs and avoid accidents for mechanical systems, e.g., wind turbine systems (Chai, An, Ma, & Sun, 2016, Zhang, Gao, Yang, & Guo, 2019), urban ropeway transport systems (Martinod, Bistorin, Castañeda, & Rezg, 2019). Most researchers focused on building perfect maintenance models in the past decades, which assumed that components after repair would return to the “as-good-as new” state. However, in practice, this assumption may not always hold due to technical limitations. Thus, the study of imperfect maintenance policies has gradually attracted more attention in recent years (Do, Voisin, Levrat, & Lung, 2015, Labeau and Segovia, 2011, Liu and Huang, 2010). Imperfect maintenance action implies that the component after repair is between the as-bad-as old state and the as-good-as new state. Additionally, in the maintenance process of the system, replacing a unit with a new one is more expensive and wasteful. Then imperfectly repairing a unit can keep the system in normal operational states for a long time, meanwhile reducing the system maintenance costs. So far, imperfect maintenance actions have been applied in many areas, e.g., the building’s envelope components (Ferreira, Silva, Brito, Dias, & Flores-Colen, 2020), and production systems (Liu, Dong, Frank Chen, Liu, & Ye, 2020).

In the literature, most researchers explored the imperfect maintenance policy based on the usage of components or systems (Yang et al., 2019). For example, according to the usage of components, (Nakagawa, 1979, Nakagawa, 1979) first attempted to develop the (p,q) rule where a perfect maintenance action occurred with probability p and an imperfect maintenance action occurred with probability 1-p. Then Brown and Proschan (1983) extended Nakagawa’s work and proposed an imperfect maintenance policy with zero repair time. Under this policy, they obtained the distribution of the interval between the two successive as-good-as new states and enriched the imperfect maintenance theory. Afterward, on the basis of Brown and Proschan’s model, Block et al. (1985) incorporated the age t into a new model which is known as the (p(t),q(t)) rule. According to the (p(t),q(t)) rule, Iyer (1992) studied the system availability with the non-zero repair time under imperfect repair. Later, the age-based imperfect maintenance policy was widely studied by related researchers (Murthy and Nguten, 1981, Shen and Griggith, 2001). Among them, Kijima et al. (1988) wrote a landmark paper in 1989, which proposed two types of imperfect maintenance models. In the Kijima model I, the imperfect maintenance action can decrease the level of system damages that only occurred during the last operating interval. In contrast, in the Kijima model II, all damages before the imperfect maintenance action can be reduced. This means that the maintenance action can return the unit to the as-good-as new state in the Kijima model II. By utilizing the lifetime distribution function of systems, Sheu et al. (2012) formulated an imperfect preventive maintenance model with improvement factors in the hazard-rate function. Martinod et al. (2018) proposed the periodic block-type imperfect maintenance policy and the age-based imperfect maintenance policy for a multi-component system to minimize the system maintenance costs. In the literature mentioned above, the optimization models of system maintenance policies were formulated mainly based on the usage of components or systems, which neglected the deterioration levels of components. Therefore, to enrich the research in this field, it is necessary to further explore the condition-based imperfect maintenance policy.

The condition-based maintenance policy has attracted more attention from researchers with the development of health monitoring techniques in recent years (Ghasemi et al., 2010, Wang et al., 2021). For example, in 2012, Do Van and Bérenguer (2012) developed different types of condition-based maintenance models by periodically monitoring a single-unit production system and further analyzed the effect of this maintenance policy on deterministic and random intervention gains. Besides, Do et al. (2015) proposed a proactive condition-based maintenance policy to incorporate perfect and imperfect maintenance actions in the process of maintenance. Based on the Kijima type Ⅱ model, Ponchet et al. (2012) proposed a virtual age model where a time-dependent improvement function was presented. Wu et al. (2017) applied the condition-based imperfect maintenance model to the power system and verified that the proposed maintenance model can effectively reduce the risk of system failure. Therefore, the condition-based imperfect maintenance policy is an effective strategy for repairmen since the condition monitor technique can effectively predict the system condition, which can effectually reduce the system shutdown risks and improve the system maintainability for repairmen. Although the condition-based maintenance strategy for a single-unit system has been widely studied, the condition-based imperfect maintenance strategy has not been thoroughly explored for non-identical components in a series system.

For an arbitrary multi-unit system, the system maintenance time is generally a random variable. However, most researchers either neglected the repair time or regarded it as a constant (Wang and Miao, 2021, Wang et al., 2021). For example, Ding and Tian, 2011, Bai and Yun, 1986, and Lin et al. (2000) formulated the imperfect maintenance models with assuming zero repair time. For short-lifetime systems, the repair time is not negligible compared with system operational time. To be more practical, the repair time of any unit in the two-unit series system is assumed to be a random variable in this paper. Moreover, the system maintenance cost is up to the repair time of units. Namely, the longer repair time will lead to more maintenance costs, whereas the shorter repair time results in lower maintenance costs. More important, an opportunistic maintenance policy is taken into consideration for a two-unit series system since a conjoined maintenance action among different units can commonly yield a lower system maintenance cost than repairing each unit separately. Thus, repairmen can take full use of the repair time of a component to simultaneously maintain the other unit with meeting the maintenance requirements. Iung, Do, Levrat, & Voisin, 2016, Hu et al., 2020 proved that the opportunistic maintenance strategy was a useful method to reduce the system maintenance costs by sharing the expensive set-up cost.

In this paper, not only age-based opportunistic maintenance action but also condition-based opportunistic maintenance action is adopted in the maintenance process of the system. An imperfect maintenance model is formulated for a two-unit series system where units 1 and 2 are subject to soft failure and hard failure, respectively. Unit 1 is regarded as subject to soft failure when its deterioration level reaches a critical threshold. For unit 2, two types of failures are considered: minor failure (failure type I) and catastrophic failure (failure type II). When type I failure with probability p occurs, a minor repair with zero repair time will be performed. When type II failure with probability 1-p is induced, unit 2 experiences serious damage, and then a major maintenance action with random repair time will be conducted on unit 2. Besides, two types of maintenance actions are considered: perfect replacement and imperfect maintenance. When the deterioration level of unit 1 or the usage of unit 2 reaches or exceeds the critical threshold, it means that the unit degradation is more serious. In this case, repairing the unit is less economical and thus the replacement action will be conducted. Otherwise, imperfect maintenance actions are performed, which can reduce the deterioration level of unit 1 and the usage of unit 2 to some degree. The semi-Markov decision process (SMDP) (Tijms, 1994) is utilized to derive the system state transition probability, the expected sojourn time, and the expected maintenance cost. The system average maintenance cost is minimized by optimizing the thresholds of the imperfect maintenance and the perfect replacement for a two-unit series system. The main contributions of this paper are summarized as follows:

  • ·

    Developing an imperfect opportunistic maintenance policy for a series system with two non-identical units.

  • ·

    Utilizing the deterioration level of unit 1 and usage of unit 2 in the maintenance process of the system.

  • ·

    Optimizing the thresholds of opportunistic maintenance and perfect replacement of two units to obtain the system minimum maintenance cost under the semi-Markov decision process.

The remainder of this paper is organized as follows. The model assumptions and maintenance policies are described in Section 2, where the objective function is presented. In Section 3, the system state transition probability, the expected system sojourn times, and the expected maintenance costs are derived in the semi-Markov decision framework. After that, an optimal solution procedure is listed in Section 4. In Section 5, an illustrative example is presented, and a comparison analysis and a sensitivity analysis are given. The conclusions and future research directions are shown in Section 6.

Section snippets

System description and assumptions

This paper considers a two-unit repairable system where unit 1 and unit 2 are subject to soft failure and hard failure, respectively. The system immediately stops working upon unit 2 failure or when the deterioration level of unit 1 reaches the critical value D at discrete inspection epochs (Δ,2Δ,), whichever occurs first. The model assumptions are as follows.

Assumption 1

The deterioration level of unit 1 can be detected at discrete inspection epochs (Δ,2Δ,). The degradation process is the Gamma process {X

Solution in the SMDP framework

First, state transition probabilities of units and imperfect maintenance costs will be derived. Second, the system state transition probabilities, the system expected sojourn times and the system expected maintenance costs will be further given. Finally, the objective function of minimizing the average system maintenance costs will be formulated according to the proposed maintenance policy.

Solution procedure

Since the decision variables are four variables (D,D1,M,M1) in this paper, it is difficult to obtain the analytical solution for the non-linear programming problem. Thus, the Cyclic Coordinate Search Algorithm (Bazaraa & Shetty, 1979) is used to solve the multi-variable optimization problem. The detailed procedure can be seen in Table 1.

An illustrative example

This paper aims to explore an optimal imperfect maintenance policy for a system with two non-identical units. In this section, an example is used to demonstrate the proposed maintenance policy for a two-unit wind turbine system. According to the history data collected from WMEP in Germany and Windstats in Denmark (Pineda et al., 2017), we can obtain the blade is subject to degradation failure. Besides, Tian, Jin, Wu, & Ding, 2011, Zhang, Gao, Yang, & Guo, 2019 pointed out that the generator was

Conclusion and future research

In this paper, a novel optimization model of age-based and condition-based imperfect maintenance policy is formulated. This paper creatively explores the imperfect maintenance policy for two non-identical components. For the series system, the two units are either perfectly replaced or imperfectly repaired based on the deterioration level of unit 1 or the usage of unit 2 at the decision epoch, respectively. Perfect replacement actions occur once the damages of the two units exceed the critical

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (51905017) which is highly appreciated by the authors.

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