A condition-based maintenance of a dependent degradation-threshold-shock model in a system with multiple degradation processes

Highlights

• A condition-based maintenance model is proposed.

• Two dependent causes of failure are considered: deterioration and external shocks.

• Deterioration is given by multiple degradation processes growing by a gamma process.

• The initiation of degradation processes follows a Non-homogeneous Poisson process.

• External shocks arrive at the system by using a Doubly Stochastic Poisson Process.

Abstract

This paper proposes a condition-based maintenance strategy for a system subject to two dependent causes of failure: degradation and sudden shocks. The internal degradation is reflected by the presence of multiple degradation processes in the system. Degradation processes start at random times following a Non-homogeneous Poisson process and their growths are modelled by using a gamma process. When the deterioration level of a degradation process exceeds a predetermined value, we assume that a degradation failure occurs. Furthermore, the system is subject to sudden shocks that arrive at the system following a Doubly Stochastic Poisson Process. A sudden shock provokes the total breakdown of the system. Thus, the state of the system is evaluated at inspection times and different maintenance tasks can be carried out. If the system is still working at an inspection time, a preventive maintenance task is performed if the deterioration level of a degradation process exceeds a certain threshold. A corrective maintenance task is performed if the system is down at an inspection time. A preventive (corrective) maintenance task implies the replacement of the system by a new one. Under this maintenance strategy, the expected cost rate function is obtained. A numerical example illustrates the analytical results.

Introduction

Maintenance refers to the set of necessary operations applied to a system so that it can work properly. Nowadays, maintenance plays an important role in most companies, which try to provide high quality products but minimizing the production cost [1]. Maintenance activities performed on an industrial system can increase not only its safety, but also ensure its availability and correct functioning. Traditionally, these maintenance tasks were allocated based on requirements in legislation, company standards or in-house maintenance experience. However, in the early 1960s, authors as Barlow and Hunter, Radner and Jorgenson or McCall [2], [3], [4] started developing mathematical models, which aim to quantify costs and to find the optimum balance between the maintenance cost on one side and the associated cost (benefit) on the other.

Maintenance strategies regulate the different maintenance tasks which will be performed on systems. In general, maintenance tasks are classified in corrective maintenance and preventive maintenance. Corrective maintenance is defined as the maintenance which is required when a system has failed. Preventative maintenance is a maintenance planned in order to prevent the occurrence of future failures. It is performed when a system is still working. Preventive maintenance activities in single-unit systems were classified by Rausand and Høyland [5] in: (i) Age-based maintenance, where maintenance tasks are performed when the system exceeds a certain age. (ii) Calendar-based maintenance, where maintenance tasks are performed at fixed time instants. (iii) Condition-based maintenance, where maintenance tasks are based on one or several variables which measure the state of the system. The state variables are monitored continuously or on a regular basis. The maintenance of the system is performed when some state-variable exceeds a certain fixed level. Because of its properties, this kind of maintenance is widely used in the current literature (see e.g. [6], [7], [8], [9]).

Many systems are degraded physically over time and a measurable physical deterioration almost always precedes the failure. The deterioration level of the system is represented by a degradation process. But, in practice, many systems are subject to multiple degradation processes due to which they can be degraded in more than one way. A characteristic example of this type of multiple degradation is the pitting corrosion. Pitting corrosion consists of the appearance of small pits on the surface of some metals and alloys. Each pit is considered as a degradation process. Pits are deteriorated in a way that when they start growing, the environment developed is such that stimulates their own growth [10]. Therefore, a pitting corrosion model is a combination of two processes: the initiation process and the growth process.

It is well established that the pitting corrosion process has a stochastic nature [11], [12]. In the current literature about degradation models, several authors studied pitting corrosion, proposing different stochastic models, both for the initiation of degradation processes and for its growth. Among others, Valor et al. [13] proposed a model where the initiation process was distributed under a Non-homogeneous Poisson process (NHPP), and the growth process was simulated using a non-homogeneous Markov process.

On the other hand, Sheikh et al. [14] treated pitting corrosion as a time-dependent stochastic damage process characterized by an exponential or logarithmic growth. Van Noortwijk and Klatter [15] developed a maintenance model for the sea-bed protection of storm-surge barriers where the stochastic initiation process and the growth process were modelled as a Poisson process and a gamma process, respectively. Mercier et al. [16] also used this combination of processes in order to model the degradation process of passive components within electric power plants. In addition to this, Kuniewski et al. [17] and Castro et al. [18] provided a new approach to pitting models by combining the NHPP for the initiation process and the gamma process for the growth.

A gamma process is a stochastic process, proposed initially by Abdel-Hameed [19] as a specific model for degradation occurring randomly in time, and characterized by independent and non-negative increments distributed by using gamma distributions with identical scale parameters. Because of its properties, the gamma process is considered as one of the most appropriate processes for the stochastic modelling of monotonic degradation accumulated over time, in a succession of tiny increments such as wear, stress, corrosion, erosion or the degradation process growth. A broad survey about gamma processes was performed by van Noortwijk [20].

However, systems are not only subject to internal degradation, but also are exposed to sudden shocks that can cause their failure. Models which consider systems subject to these two competing causes of failure (degradation and shocks) are called Degradation-Threshold-Shock (DTS) models. As far as we know, Lemoine and Wenocur [21] were the first to combine both competing causes of failure. So, the failure time of a system subject to degradation and shocks is the minimum of the moment when degradation first reaches a critical threshold and the moment when a shock occurs. Lin et al. [22] proposed a DTS model where the growth process followed a continuous-time semi-Markov process and where the process of sudden shocks was distributed by using a Homogeneous Poisson process (HPP). Wang et al. [23] analysed a DTS model where the growth of the degradation process followed a Gaussian process and the occurrence of shocks was considered both as a fixed time period and as varying time periods under an HPP. Singpurwalla [24] and Cox [25] developed different approaches to analyse stochastic process-based models, including also DTS models.

Some authors analysed maintenance strategies applied to DTS models subject to only one degradation process, Li and Pham [26] provided a condition-based maintenance of a system where degradation was modelled as a function of time and the process of shocks was distributed by using a compound Poisson process. Huynh et al. [27], [28] modelled different maintenance strategies for systems where the growth of the degradation process was distributed by a gamma process and the process of sudden shocks followed an NHPP with intensity dependent on the deterioration level of the system. Asadzadeh and Azadeh [29] proposed a condition-based maintenance model describing fatigue crack degradation with a Paris–Erdogan model and random positive normal shocks with Weibull frequency model. XiaoFei and Min [30] analysed a system subject to shocks arriving following a time HPP and degradation process distributed under a typical Poisson stochastic distribution. Deloux et al. [31] combined statistical process control and condition-based maintenance for a system with two failure mechanisms, deterioration and shocks due to a stressful environment. They propose to model the failure process by two explicative variables a cumulative deterioration process, which following a non-decreasing stochastic process and a stress covariate, distributed under a stochastic process fluctuating around a given mean.

On the other hand, DTS models where systems subject to only one degradation process are extended by different authors, which analyse systems subject to multiple degradation processes. Castro [32] and Castro et al. [33] studied different maintenance strategies for systems subject to multiple degradation processes and sudden shocks. They assumed that both causes of failure (degradation and sudden shocks) are independent. Li and Pham [34] proposed a stochastic failure model for a system subject to two degradation processes and to a shock process. They suppose that the growth process was distributed as different continuous probabilistic functions and shocks under a compound Poisson process. Wang and Pham [35] provided a model for a system subject to m degradation processes described by a function of time influenced by the shock process, distributed by using an HPP.

In this paper, we propose a condition-based maintenance strategy for a system subject to multiple degradation processes and sudden shocks, assuming that both causes of failure are dependent. Degradation processes start at random times following an NHPP and grow according to a gamma process. The system fails when the deterioration level of a degradation process exceeds a certain threshold. In addition, sudden shocks arrive at the system according to a Doubly Stochastic Poisson Process (DSPP), also called Cox process. This kind of processes describes a Poisson process with a random intensity that might represent a random environment or a field that influences the Poisson locations of the points. They were first introduced by Cox [36] and later were described by Cox and Lewis [37]. Afterwards, different authors considered the shock process as a DSPP (see e.g. [38], [39]). In our model, the deterioration levels of the existing degradation processes in the system influence the sudden shock process. A sudden shock provokes the total failure of the system.

Since the continuous monitoring of the system is too costly, it is inspected each T time units. These instants are called inspection times. During inspections, the state of the system and the deterioration level of each degradation process is checked. If the system has not failed, the deterioration level of each existing degradation process in the system is measured. Let M be a certain threshold from which we consider the system is still functioning but is too deteriorated. A preventive maintenance task is performed when the deterioration level of a degradation process has exceeded this predetermined deterioration level M in an inspection time. A corrective maintenance task is performed when the system is down in the inspection time.

Each maintenance task has a fixed cost associated. An objective of this paper is to find optimal values both for the threshold M and for the time between inspections T. This implies to obtain values for M and T which minimize an objective cost function. Let $\left\{Z\right(t),t\ge 0\}$ be a regenerative process. We denote by $D_1,D_2,D_3,\dots$ their regeneration epochs, by $L_i=D_i-D_{i-1}$, $i=1,2,\dots$ the length of the i-th renewal cycle being $D_0=0$, by C(t) the cost of the system at time t, and by C_i the total cost in the i-th renewal cycle. Therefore, if a cost structure is imposed on the regenerative process $\left\{Z\right(t),t\ge 0\}$, it is well known that (see [40, p. 41])

$$\frac{C\left(t\right)}{t}\to \frac{E\left[C_1\right]}{E\left[L_1\right]},$$

with probability one, that means, the long-run average cost per time unit is equal to the expected cost in a renewal cycle divided by the expected length of this cycle for almost any realization of the process. The long-run average cost criterion is widely used in the reliability literature [28], [27] and we shall use (1) as objective function to analyse the optimal strategy for this maintenance model.

This paper is structured as follows. In Section 2 the general framework of the model is described. Section 3 details the maintenance strategy used in this model. An illustrative example is shown in Section 4. Section 5 concludes and shows further possible extensions of this paper.