Tuan Khac

Huynh

K T Huynh

An adaptive predictive maintenance model for repairable deteriorating systems using inverse Gaussian degradation process

Keywords: Adaptive maintenance decision, Imperfect repair, Inverse Gaussian process, Predictive maintenance, Remaining useful life, Semi-regenerative process

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

system deterioration level at time t ≥ 0, system degradation process IGP (•, •), IG (•, •), IG process, IG distribution E j , S j end time of the j-th repair/replacement, starting time of the ( j + 1)-st repair/replacement µ 0 , µ 1 x j constant and x j -dependent parts of the shape parameter µ x j of IG process {X t } E + j ≤t≤S j with X E + j = x j λ constant scale parameter of {X t } t≥0 Φ (•), ϕ (•), standard normal cdf and pdf T j,k k-th inspection time during the ( j + 1)-st repair/replacement cycle E + j , E + j+1 f (•; •, •), F (•; •, •), F (•; •, •) pdf, cdf and sf of IG probability law L, R T j,k system failure threshold, system RUL at T j,k FR T j,k expectation with respect to the stationary measure π

1 {•} , δ 0 (•)
indicator function, Dirac mass at 0

Introduction

With usage, age and environmental impacts, many industrial systems suffer continuous degradation leading eventually to random failure, no matter how good they are designed. Cutting tools are subject to cumulative wear [START_REF] Benkedjouh | Health assessment and life prediction of cutting tools based on support vector regression[END_REF], hydrocarbon pipelines undergo corrosion [START_REF] Amaya-Gómez | Modeling of pipeline corrosion degradation mechanism with a lévy process based on ili (in-line) inspections[END_REF], hydraulic structures suffer erosion [START_REF] Zarrabi | Effect of hydraulic parameters on abrasion erosion of fiber reinforced concrete in hydraulic structures[END_REF], water-feeding turbo-pumps of nuclear power plants endure fatigue crack growth [START_REF] Baraldi | Modelling the effects of maintenance on the degradation of a water-feeding turbo-pump of a nuclear power plant[END_REF], etc. The failure of such systems causes damage not only to the industry, but also to the society and the environment. Therefore, keeping their efficient operation at high reliability is vital to enterprises, especially in the context of aggressive global competition.

When the systems are repairable, various maintenance actions, such as inspection, testing, repair, replacement, etc., are conceivable [START_REF] Ahuja | Total productive maintenance: literature review and directions[END_REF]. Nevertheless, these actions are themselves time-consuming and costly, seeking an effective maintenance solution to gain a competitive advantage is thus of great concern to enterprises. Among existing solutions (see e.g., [START_REF] De Jonge | A review on maintenance optimization[END_REF] for a recent overview), predictive maintenance (PdM) could be most appropriate [START_REF] De Jonge | The influence of practical factors on the benefits of condition-based maintenance over time-based maintenance[END_REF]. It uses collected condition monitoring data to predict the future system health in real-time, and through that knowledge, enables maintenance decision-making [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. As such, the PdM allows to save resources by carrying out proper and timely actions only when necessary. This definite advantage, together with the dissemination of computer-based monitoring technologies, has promoted the rapid development of PdM models for continuously deteriorating systems during the last two decades [START_REF] Alaswad | A review on condition-based maintenance optimization models for stochastically deteriorating system[END_REF]. Basically, the PdM modeling includes four connected steps [START_REF] Bérenguer | On the mathematical condition-based maintenance modelling for continuously deteriorating systems[END_REF]: (i) continuous degradation modeling, (ii) maintenance effects modeling, (iii) maintenance policy elaboration, and (iv) performance assessment.

State-of-the-art

Continuous degradation modeling is critical for system health prognostics. In the current state-of-the-art, it relies mostly on Lévy or diffusion stochastic processes that fit in with condition monitoring data [START_REF] Nguyen | Model selection for degradation modeling and prognosis with health monitoring data[END_REF]. The Gamma process is the most common choice for monotonic degradation because of its physical meaning and mathematical tractability [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]. When this choice fails (e.g., in the case of GaAs laser degradation data [START_REF] Wang | An inverse gaussian process model for degradation data[END_REF] or energy pipeline corrosion data [START_REF] Qin | Inverse gaussian process-based corrosion growth modeling and its application in the reliability analysis for energy pipelines[END_REF]), the inverse Gaussian (IG) process [START_REF] Wasan | On an inverse gaussian process[END_REF] could be a good alternative. Indeed, it is proved more flexible than the Gamma process in incorporating random effects and covariates, while still holding similar physical and mathematical meanings [START_REF] Ye | The inverse gaussian process as a degradation model[END_REF]. This explains why the IG process has recently attracted considerable attentions, especially in accelerated degradation test optimization [START_REF] Ye | Accelerated degradation test planning using the inverse gaussian process[END_REF] and in the system remaining useful life (RUL) assessment [START_REF] Guo | Improved inverse gaussian process and bootstrap: Degradation and reliability metrics[END_REF].

However, up to now, its applications in PdM modeling are still very scarce despite its versatility [START_REF] Chen | Condition-based maintenance using the inverse gaussian degradation model[END_REF].

Maintenance effects modeling describes the impacts of maintenance actions on the system degradation behavior. Perfect maintenance with as-good-as-new (AGAN) effect is certainly the best-known assumption in the literature [START_REF] Huynh | A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events[END_REF]. However, it cannot cover numerous realistic actions whose imperfectness may be induced by influential factors such as human errors, spare parts quality, lack of materials, lack of maintenance time, etc. To meet this practical need, imperfect maintenance (IM) models (see [START_REF] Pham | Imperfect maintenance[END_REF] for a complete overview) have been extensively studied under the assumption that the system condition after a maintenance is worse-than-new but better-than-old. Despite those efforts, modeling IM effects in the context of degradation processes is still not active, and mainly mimics the ideas of lifetime-based IM models (see e.g., [START_REF] Ponchet | Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements[END_REF][START_REF] Mercier | On the modelling of imperfect repairs for a continuously monitored gamma wear process through age reduction[END_REF][START_REF] Zhang | Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance[END_REF][START_REF] Kahle | Imperfect repair in degradation processes: A kijima-type approach[END_REF]). Furthermore, in most existing models, past and present IM actions are linked by the memory assumption [START_REF] Doyen | Classes of imperfect repair models based on reduction of failure intensity or virtual age[END_REF]. This assumption is however not easily verified in practice owing to the stochastic nature of degradation processes. How to break this strong assumption to reach more feasible past-dependent IM models remains an open issue.

PdM policy specifies actions to be adopted at decision epochs based on the system health evolution. In the literature, the elaboration of such a policy has mainly focused on inspection schedules and thresholds for preventive maintenance decision-making [START_REF] Ahmad | A review of condition-based maintenance decision-making[END_REF].

A policy with periodic inspection and fixed degradation-based threshold is perhaps the simplest one (see e.g., [START_REF] Lee | Optimal replacement strategy for stochastic deteriorating system with random wear limit under periodic inspections[END_REF][START_REF] Huynh | A condition-based maintenance model with past-dependent imperfect preventive repairs for continuously deteriorating systems[END_REF]). This simplification allows easy applications in practice, but also implies a non-optimal maintenance policy. Therefore, much attentions have been paid to make the policy more flexible seeking higher performance. Some authors suggested non-periodic inspection schemes by adapting the inter-inspection intervals to the system age [START_REF] Golmakani | Optimal age-based inspection scheme for condition-based maintenance using a* search algorithm[END_REF], system degradation level [START_REF] Grall | A condition-based maintenance policy for stochastically deteriorating systems[END_REF], system degradation rate [START_REF] Ponchet | Assessment of a maintenance model for a multi-deteriorating mode system[END_REF], system RUL [START_REF] Omshi | A dynamic auto-adaptive predictive maintenance policy for degradation with unknown parameters[END_REF], or to the working environment [START_REF] Huynh | Adaptive condition-based maintenance decision framework for deteriorating systems operating under variable environment and uncertain condition monitoring[END_REF]. Meanwhile, some others proposed using varied preventive maintenance thresholds (see e.g., [START_REF] Chen | Condition-based maintenance using the inverse gaussian degradation model[END_REF][START_REF] Wu | A dynamic condition-based maintenance model using inverse gaussian process[END_REF]). These solutions usually lead to very complex, even unfeasible, maintenance policies because of a high number of decision variables. So, a big challenge is to build an adaptive PdM policy, on the one hand, simple enough to facilitate practical applications, and on the other hand, sophisticated enough to fulfill certain performance criteria (e.g., cost, availability, etc.). The ultimate goal of performance assessment is to optimize maintenance policies by adjusting their decision variables (interinspection intervals, maintenance thresholds, etc.) so that the maintenance performance is maximum. As far as economic criteria and mathematical models are concerned, the performance assessment can be done by two main approaches [START_REF] Frangopol | Probabilistic models for life-cycle performance of deteriorating structures: review and future directions[END_REF]. The first approach relies on the (semi)-Markov decision process and dynamic programming or policy iteration tools [START_REF] Puterman | Markov Decision Processes: Discrete Stochastic Dynamic Programming[END_REF]. It requires transforming the continuous-state space of degradation processes into the associated discrete-state space [START_REF] Chen | Condition-based maintenance using the inverse gaussian degradation model[END_REF][START_REF] Nguyen | Joint optimization of monitoring quality and replacement decisions in condition-based maintenance[END_REF]. Note, however, that such a discretization could be undesirable in some cases where the intrinsic continuity of degradation processes is more significant for maintenance decision-makers [START_REF] Bérenguer | On the mathematical condition-based maintenance modelling for continuously deteriorating systems[END_REF]. The second approach allows avoiding this obstacle. Indeed, using the (semi)-regenerative theory or (Markov)-renewal theory [START_REF] ¸inlar | Markov renewal theory: A survey[END_REF], this approach can be effectively applied for degradation processes with continuous-state space [START_REF] Grall | Continuous-time predictive-maintenance scheduling for a deteriorating system[END_REF],

as well as with discrete-state space [START_REF] De Jonge | Discretizing continuous-time continuous-state deterioration processes, with an application to condition-based maintenance optimization[END_REF]. More importantly, it results in full analytical cost models rather than numerical solutions given by the first approach. Recently, several authors have used the semi-regenerative technique to evaluate and optimize the long-run maintenance cost rate of static maintenance policies (see e.g., [START_REF] Sun | Managing component degradation in series systems for balancing degradation through reallocation and maintenance[END_REF][START_REF] Hu | Opportunistic maintenance for two-component series systems subject to dependent degradation and shock[END_REF]). This technique is even more meaningful in the context of adaptive PdM policies [START_REF] Grall | Continuous-time predictive-maintenance scheduling for a deteriorating system[END_REF][START_REF] Castanier | A sequential condition-based repair/replacement policy with non-periodic inspections for a system subject to continuous wear[END_REF], but more effort in mathematical formulation is, no doubt, required [START_REF] Huynh | Multi-level decision-making for the predictive maintenance of k-out-of-n:f deteriorating systems[END_REF].

Contributions and organization

In this paper, our ambition is to seek answers to the above-mentioned shortcomings of the PdM modeling. Consequently, a new adaptive PdM model is developed for a continuously deteriorating single-unit system subject to periodic inspection, imperfect repair (IR) and perfect replacement. The inspection and replacement are assumed memoryless, while the repair is past-dependent in the sense that it cannot bring the system back to a degradation level better than the one reached at the last repair [START_REF] Huynh | A condition-based maintenance model with past-dependent imperfect preventive repairs for continuously deteriorating systems[END_REF]. To characterize such an imperfect effect, we sample the system degradation level after a repair from a probability distribution truncated by the degradation levels just before the current repair and just after the last repair/replacement. This allows breaking the memory assumption usually made in the IM modeling (see e.g., arithmetic reduction model [START_REF] Ponchet | Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements[END_REF] or Kijima's type model [START_REF] Kahle | Imperfect repair in degradation processes: A kijima-type approach[END_REF]). We also take into account the impact of maintenance efficiency on the system degradation behavior via the random effect of an IG degradation process. Indeed, expressing its shape parameter as an increasing function of the degradation level returned by a system repair/replacement, we consider that the system degradation is faster and more chaotic when the maintenance efficiency is lower. Some real-world examples of this phenomenon are the degradation paths of gyroscopes and draught fans provided respectively in [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF] and [START_REF] Wang | Remaining useful life prediction of degrading systems subjected to imperfect maintenance: Application to draught fans[END_REF]. Besides, the considered maintenance actions consume different duration and incur different costs. All of them should be properly coordinated into a unifying policy to enable the best maintenance performance. Using the system RUL and the maintenance duration as decision variables, we build an adaptive PdM policy that allows (i) to determine the right moment for a switch from periodic inspection to a maintenance, (ii) to dynamically schedule maintenance times, and (iii) to select the suitable maintenance action (i.e., either IR or perfect replacement) at a scheduled time. Interestingly, the versatility of decision variables offers a good compromise between simplicity and performance. To assess the economic performance of the policy in the long-run, we analyze the probabilistic behavior of the maintained system at steady state, and analytically evaluate its maintenance cost rate using the semi-regenerative technique. The main difficulty lies in the difference in the nature of decision variables. We thus transform all these variables into associated degradation levels to overcome this obstacle. Various numerical experiments and comparative studies with benchmarks under different system configurations allow to illustrate the developed adaptive PdM model, and to confirm its cost-effectiveness.

The paper remainder is organized as follows. Section 2 describes the considered maintained system. Its probabilistic behavior at steady state is analyzed in Section 3. Sections 4 is devoted to the cost model formulation and optimization for the maintained system. Section 5 gives more insight into the effectiveness of the proposed adaptive PdM model thanks to numerical assessments. Finally, conclusions and perspectives are discussed in Section 6.

Description of the maintained system

Let us consider a continuously deteriorating single-unit system subject to inspection, imperfect repair and perfect replacement.

Its health state at time t ≥ 0 is completely characterized by a scalar random variable X t ≥ 0, with x j = 0. In the absence of repair or replacement, the system evolves from its new state (i.e., X t = 0) to its failure state (i.e., X t ≥ L) following an increasing stochastic degradation process {X t } t≥0 . Each of maintenance actions has its own effects on the behavior of {X t } t≥0 and incurs different cost. From an economic viewpoint, they should be coordinated into a unified adaptive PdM policy. The aim of this section is to model such a maintained system based on the assumption that {X t } t≥0 acts like an inverse Gaussian process. To facilitate the comprehension, the following description focuses on the ( j + 1)-st repair/replacement cycle which is the time interval between two successive end-ofrepair/replacement times E j and E j+1 , with E 0 = 0 and j ∈ N.

Degradation modeling

Let S j be the starting time of a repair/replacement within the cycle E + j , E + j+1 , the system degradation over E + j , S j evolves from X E + j = x j following a stationary IG process with shape parameter µ x j and scale parameter λ > 0

{X t } E + j ≤t<S j ∼ IGP µ x j , λ . (1)
The shape function µ x j is the sum of two elements: (i) a constant µ 0 > 0 characterizing the proper dynamics of the system degradation, and (ii) a function µ 1 x j ≥ 0 representing the effects of repair/replacement actions before E j on the dynamics of {X t } E + j ≤t<S j via the recovery degradation level

x j µ x j = µ 0 + µ 1 x j . (2)
As in [START_REF] Huynh | Modeling past-dependent partial repairs for condition-based maintenance of continuously deteriorating systems[END_REF], we shall express µ 1 x j as an increasing function of x j with µ 1 (0) = 0 to take into account the phenomenon that the lower the efficiency of past repairs, the more the system is chaotic and vulnerable to degradation. Therefore, the degradation increment between two times s and t, with E + j ≤ s < t ≤ S j , is IG distributed with shape parameter µ x j • (ts) and scale parameter λ 2 . Its probability density function (pdf), cumulative distribution function (cdf) and survival function (sf) are respectively [START_REF] Ahuja | Total productive maintenance: literature review and directions[END_REF] where 1 {•} is the indicator function which equals 1 if the argument is true and 0 otherwise, Φ (•) denotes the standard normal cdf

• (t -s) 2 , i.e., X t -X s ∼ IG µ x j • (t -s) , λ • (t -s)
Φ (x) = 1 √ 2π x -∞ exp - t 2 2 dt. ( 6
)
These probability functions are the basis to develop the stationary law and the cost model of the maintained system (see Sections 3 and 4). The IG degradation process IGP µ x j , λ has the mean rate µ x j and the variance rate µ 3 x j /λ over E + j , E + j+1 . Despite its stationarity over a given repair/replacement cycle, this process is non-stationary from cycle to cycle owing to the randomness of x j , j ∈ N.

Inspection and remaining useful life estimation

The inspection is merely an information-taking operation that reveals perfectly the current hidden degradation level of the system.

We assume that the inspection takes negligible time and has no effect on the system degradation behavior. This is why the system degradation level just before and just after a k-th inspection time, k ∈ N * , during the cycle E + j , E + j+1 remains unchanged (i.e., X T - j,k = X T + j,k = X T j,k ). Given X T j,k , we can further access the system RUL using the degradation model IGP µ x j , λ . Indeed, let L denote a fixed failure threshold, if no more maintenance action is planned, the system RUL at T j,k given X T j,k = x j,k < L and X E + j = x j < x j,k is defined by the random variable

R T j,k = inf r > 0 : X T j,k +r ≥ L | X T j,k = x j,k , X E + j = x j . (7)
We are interested in the sf and the standard deviation (std) of R T j,k which are measures of the predictive system reliability and of the prognosis precision respectively. The increasing of IGP µ x j , λ allows to compute the sf of R T j,k at r as

FR T j,k |X T j,k ,X E + j r | x j,k , x j = P X T j,k +r < L | X T j,k = x j,k , X E + j = x j = F L -x j,k ; µ x j • r, λ • r 2 , (8)
where F X s+r -X s (•) is given by (4). Meanwhile, the std of R T j,k can be obtained from FR T j,k |X T j,k ,X E + j r | x j,k , x j as follows

S R T j,k | X T j,k = x j,k , X E + j = x j = µ x j λ 1 4 + β 2 β 2 + 3 Φ (β) + β β 2 + 2 ϕ (β) -β 2 + 1 Φ (β) + βϕ (β) 2 , (9)
where

β ≔ β x j,k , x j = λ • L -x j,k /µ x j . We remark that both FR T j,k |X T j,k ,X E + j r | x j,k , x j and S R T j,k | X T j,k = x j,k , X E + j =
x j are independent of the inspection time T j,k , and always decreasing with respect to x j,k or x j (see also Subsection 2.6). This property makes these measures more versatile for predictive maintenance than the degradation levels X T j,k and X E + j . We shall use S R T j,k | X T j,k = x j,k , X E + j = x j to decide a switching from an inspection to a maintenance at an inspection time T j,k , and FR T j,k |X T j,k ,X E + j r | x j,k , x j for maintenance planning because of the time mark r.

Replacement and repair

The replacement is a perfect action that always makes the system AGAN independently of previous system states and interventions. It takes a fixed duration ρ 0 due to, e.g., the material set-up, the system dismantling and reassembly, etc. During the replacement duration, the system is assumed inactivated. Therefore, if a replacement starts at S j , it will end at E j+1 = S j + ρ 0 . The corresponding system degradation levels are X S j = X E - j+1 and X E + j+1 = 0. The repair, on the contrary, is an IM action whose duration and efficiency depend closely on the system degradation levels returned by the last replacement/repair X E + j and at the beginning of the current repair X S j . More concretely, the repair duration includes not only a fixed duration ρ 0 due to the same reasons as in the replacement, but also a variable duration ρ 1 X E + j , X S j depending on X S j and X

E + j ρ X E + j , X S j = ρ 0 + ρ 1 X E + j , X S j , (10)
with ρ 1 (0, 0) = 0. Especially, ρ 1 X E + j , X S j ≥ 0 is an increasing function of X E + j and X S + j , because a higher degraded system (i.e., higher X S j ) which has been undergone more repairs since the last renewal (hence higher X E + j ) requires naturally a longer repair duration [START_REF] Castanier | A sequential condition-based repair/replacement policy with non-periodic inspections for a system subject to continuous wear[END_REF]. Throughout the repair, the system is inactivated:

X S j = X E - j+1 with E j+1 = S j + ρ X E + j , X S j .
Once the repair ends, the system degradation is returned to a random level between X E + j and X S j (i.e., X E + j ≤ X E + j+1 ≤ X S j ). Such a condition implies the impact of previous maintenance actions on the current repair (i.e., X E + j ≤ X E + j+1 ) while retaining the inherent better-than-old characteristic (i.e., X E + j+1 ≤ X S + j ) [START_REF] Pham | Imperfect maintenance[END_REF]. Evidently, a system, which has been repaired several times since the last renewal, is subject to lower efficiency even the same repair quality [START_REF] Huynh | A condition-based maintenance model with past-dependent imperfect preventive repairs for continuously deteriorating systems[END_REF][START_REF] Liao | Maintenance of continuously monitored degrading systems[END_REF][START_REF] Castro | Performance measures for a deteriorating system subject to imperfect maintenance and delayed repairs[END_REF]. To model such a repair efficiency, we represent X E + j+1 as a random sample of a truncated pdf with lower bound X E + j and upper bound X E - j+1 [START_REF] Huynh | A hybrid condition-based maintenance model for deteriorating systems subject to nonmemoryless imperfect repairs and perfect replacements[END_REF]]

X E + j+1 ∼ g x j+1 | X E + j , X S j . (11)
This model allows avoiding the strong memory assumption usually made in the IM modeling: the system after a repair is put back to an exact deterioration level where it was in the past (see e.g., [START_REF] Ponchet | Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements[END_REF] and [START_REF] Kahle | Imperfect repair in degradation processes: A kijima-type approach[END_REF]).

Maintenance costs

We denote C m , C r and C o the constant unit costs for inspection, repair and replacement respectively. C r and C o include already the inspection cost. The effects of maintenance actions on the system degradation imply the relationship

C o > C r > C m > 0.
Besides, depending on the system state at a maintenance time, additional costs could be considered. During a preventive maintenance, a running system is inactivated at a constant cost rate C i . Whereas, a failed system induces an unavailability cost at constant rate C u from the failure time to the end of the next corrective maintenance (i.e., also including the required duration for the corrective maintenance). Since the system unavailability is unforeseen, it incurs a higher cost rate than the system inactivity (i.e., C u > C i > 0). Therefore, the cumulative maintenance cost incurred in the time interval [0, t] can be expressed as

C (t) = C m • N m (t) + C r • N r (t) + C o • N o (t) + C i • I (t) + C u • U (t) , (12)
where N m (t), N r (t) and N o (t) denote respectively the number of inspections, of repairs and of replacements up to time t, I (t) and U (t) are the total duration of system inactivity and unavailability in [0, t]. Our aim is to build an adaptive PdM policy allowing the lowest long-run maintenance cost rate [52, chapter 11]

C ∞ = lim t→∞ C (t) t , (13)
with C (t) given from [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]. Note that C m , C r and C o are given in cost unit, C i , C u and C ∞ are given in cost unit/time unit, and I (t), U (t) are given in time unit. The cost unit may be U.S. Dollar, Euro, British Pound, etc., while time unit may be month, quarter, year, etc. The cost unit may be U.S. Dollar, Euro, British Pound, etc., and time unit may be month, quarter, year, etc.

Policy structure

Seeking the lowest C ∞ leads to two questions: (i) when should we intervene on the system? and (ii) what should we do at an intervention time? Following the PdM concept, we rely on the std of RUL to decide, at a periodic inspection time, whether to continue inspection or to switch to maintenance. If the latter is chosen, we next use the sf of RUL to determine a proper maintenance time. The nature of maintenance operation (i.e., repair or replacement) will depend on associated maintenance duration. As in [START_REF] Huynh | A parametric predictive maintenance decision-making framework considering improved system health prognosis precision[END_REF], we prefer using a parametric structure (with decision variables δ, σ, α and τ) to build the desired PdM policy. Consequently, the maintenance plans and decision rules over the ( j + 1)-st maintenance cycle E + j , E + j+1 , with X E + j , are schematized in Figure 1.

start The system is regularly inspected at times T j,k = E j + k • δ, with k = 0, 1, 2, . . ., and inspection period δ > 0. Note that the moment T j,0 = E j of the cycle E + j , E + j+1 is just a "fictional" inspection time, at which the system degradation level has already known by the last repair/replacement X T + j,0 = X E + j . This is why no cost is counted at T j,0 = E j . Given X T j,k , the following rules are adopted. 1. If X T j,k ≥ L, the system has been failed within the interval T j,k-1 , T j,k , and a corrective replacement (CR) starts immediately at T j,k . After the CR, the system is AGAN (i.e.,

E j , X E + j T j,k ← E j S R T j,k | X T j,k , X E + j ≥ σ? inspection at T j,k ← T j,k + δ X T j,k ≥ L? CR starts at T j,k E j+1 ← T j,k + ρ 0 X E + j+1 ← 0 stop maintenance plan at S j ← T j,k + ψ α, X T j,k , X E + j X S j ≥ L? CR starts at S j E j+1 ← S j + ρ 0 X E + j+1 ← 0 PR starts at S j E j+1 ← S j + ρ 0 X E + j+1 ← 0 ρ X E + j , X S j ≥ τ? IR starts at S j E j+1 ← S j + ρ X E + j , X S j X E + j+1 ∼ g y | X E + j , X S j
X E + j+1 = 0 with E j+1 = T j,k + ρ 0 ). 2. If X T j,k < L and S R T j,k | X T j,k , X E + j
< σ, the system is still running at T j,k and its RUL can be predicted with an acceptable precision. No further inspection is thus needed, and the next maintenance is planned at

S j = T j,k + ψ α, X T j,k , X E + j with ψ α, X T j,k , X E + j = r ≥ 0 | FR T j,k |X T j,k ,X E + j r | x j,k , x j = α (14)
and 0 ≤ α ≤ 1. The decrease of FR T j,k |X T j,k ,X E + j r | x j,k , x j allows dynamic maintenance times adaptive to the degradation levels X E + j and X T j,k while keeping the system reliability at a desired level α. The choice of a maintenance action depends on the degradation level X S j and on the associated maintenance duration.

(a) If X S j ≥ L, the system has been failed within the interval T j,k , S j , a CR with duration ρ 0 is triggered immediately at S j to reset the system to an AGAN state (i.e., X E + j+1 = 0 with E j+1 = S j + ρ 0 ). (b) If X S j < L and ρ X E + j , X S j ≥ τ, the system is still running at S j , and the "long" duration ρ X E + j , X S j implies that an imperfect repair (IR) is not appropriate for the current maintenance. This is why a preventive replacement (PR) should be carried out immediately at S j . After the PR, the system is AGAN (i.e., X E + j+1 = 0 with E j+1 = S j + ρ 0 ). (c) If X S j < L and ρ X E + j , X S j < τ, we perform an IR immediately at S j on the running system because of its economic efficiency. After the preventive IR, the system degradation X E + j+1 returns to a random level between X E + j and X S j such that

X E + j+1 ∼ g y | X E + j , X S j , where E j+1 = S j + ρ X E + j , X S j and g is a known truncated pdf. 3. If X T j,k < L and S R T j,k | X T j,k , X E + j ≥ σ,
additional inspections are needed to reinforce the precision of RUL prediction.

Accordingly, the decisions is postponed to the next inspection time at T j,k+1 = T j,k + δ.

The next maintenance cycle begins at E +

j+1 with initial deterioration level X E + j+1 . For this maintenance policy, the decision variables δ, σ, α and τ are parameters to be optimized. To highlight their importance, we call the policy (δ, σ, α, τ).

Numerical illustration

For a numerical illustration, we consider a maintained single-unit system defined by L = 15, µ

X E + j = 1 + 0.1 • X E + j , λ = 4, ρ X E + j , X S j = ρ 0 + ρ 1 X E + j , X S j = 1 + 0.1 • X E + j + 0.2 • X S j , a continuous uniform pdf for g x j+1 | X E + j , X S j , δ = 3, σ = 1.1, α = 0.95 and τ = 4. We first sketch in Figures 2a and 2b the shapes of FR T j,k |X T j,k ,X E + j r | x j,k , x j and S R T j,k | X T j,k = x j,k , X E + j =
x j respectively. Obviously, they are decreasing with respect to x j,k and x j , hence easily controllable and more informative than the sole

(b) S R T j,k | X T j,k = x j,k , X E + j = x j
Figure 2: Shapes of the sf and std of the system RUL X T j,k or X E + j . We next represent the schematic behavior of the maintained system and the associated maintenance actions in Figure 3. For a specific explanation, let us describe what happens on the system during the fourth maintenance cycle E + 3 , E + 4 (see Figure 3a), the system behavior and the maintenance actions over other cycles are similar. As shown in Figure 3b, the std of the system RUL at the three first inspection times of the cycle is still greater than the threshold σ, so further inspections with period δ are needed.

At the fourth inspection time T 3,4 , the RUL std S R T 3,4 | X T 3,4 , X E + 3 < σ, no more inspection is thus required, and a maintenance action is planned ψ α, X T 3,4 , X E + 3 time unit later as depicted in Figure 3c. At the maintenance time S 3 = T 3,4 + ψ α, X T 3,4 , X E + 3 , the system degradation level is still smaller than the failure threshold L (see Figure 3a), so a preventive action is immediately carried out on the running system. Since the duration required for a repair ρ X E + 3 , X S 3 is less than the threshold τ, the maintenance action is an IR instead of a PR. During the IR, the system is inactivated. After ρ X E + 3 , X S 3 time unit, the system degradation is returned to a level X E + 4 ∼ g x 4 | X E + 3 , X S 3 , and the maintenance cycle ends. We can find that the fixed thresholds σ and τ in Figures 3b and3d correspond to variable degradation thresholds x σ and x τ in Figure 3a (see also Subsection 3.1 for the definition of x σ and x τ ). Besides, a single threshold α can match with various waiting times depending on the system degradation state (see Figure 3c). This high flexibility confirms the versatility of the (δ, σ, α, τ) policy without a large number of decision variables.

{X t } t≥0 t L x σ x τ inspection replacement repair ψ(α, X T3,4 , X E + 3 ) ρ(X E + 3 , X S3 ) E 3 E 4 T 3,4
S(R Tj,k | X Tj,k , X E + j ) T j,k σ S(R T3,4 | X T3,4 , X E + 3 ) T 3,4 (
ψ(α, X Tj,k , X E + j ) T j,k ψ(α, X T3,4 , X E + 3 ) T 3,4 (

Probabilistic behavior of the maintained system at steady state

Evaluation of the cost rate C ∞ requires the knowledge about the probabilistic degradation behavior of the maintained system at steady state [START_REF] Bérenguer | On the mathematical condition-based maintenance modelling for continuously deteriorating systems[END_REF]. In this section, we take advantage of the semi-regenerative properties of the degradation process {X t } t≥0 to derive such a stationary measure. Indeed, given X E + j returned by a repair or replacement at time E j , j ∈ N, the future {X t } t>E j of the degradation process {X t } t≥0 is conditionally independent of its past {X t } 0≤t<E j . Following [START_REF] ¸inlar | Periodicity in markov renewal theory[END_REF], {X t } t≥0 appears as a semi-regenerative process with an embedded Markov renewal process Y j , E +

π dx j+1 = [0,L) P x j , dx j+1 π dx j , (15)
where P x j , dx j+1 stands for the transition pdf of Y j j∈N from X E + j = x j to X E + j+1 = x j+1 . We seek hereinafter a closed-form expression of P x j , dx j+1 , and thence a solution for (15).

Transition probability density function P x j , dx j+1

The main difficulty in formulating P x j , dx j+1 is the difference in the nature of the decision thresholds α and τ. To overcome this obstacle, we first convert all these thresholds into the corresponding degradation levels x σ x j and x τ x j such that

x σ x j = inf x j ≤ x j,k ≤ L : S R T j,k | X T j,k = x j,k , X E + j = x j ≤ σ , (16)
and

x τ x j = min L, inf x s ≥ x j : ρ X E + j = x j , X S j = x s ≥ τ , (17)
where S R T j,k | X T j,k = x j,k , X E + j = x j and ρ X E + j = x j , X S j = x s are given from ( 9) and ( 10) respectively. Next, we exhaustively analyze all maintenance scenarios over E + j , E + j+1 based on the system degradation, and effectuate associated probabilistic computations. Consequently, we obtain the closed-form expression of P x j , dx j+1 as

P x j , dx j+1 = δ 0 dx j+1 • p x j + f x j+1 | x j • dx j+1 , (18)
where δ 0 (•) stands for the Dirac mass at 0, p x j denotes the probability that the system is AGAN due to a PR or CR

p x j = 1 {xτ(xj)≤xσ(xj)} • 1 {0≤xj≤L} + p 1 x j + p 2 x j + p 3 x j + p 4 x j • 1 {0≤xj<xσ(xj)<xτ(xj)} + p 5 x j • 1 {xσ(xj)≤xj<xτ(xj)} + 1 {xσ(xj)<xτ(xj)≤xj<L} , (19)
and f x j+1 | x j is the pdf of the system degradation state after an IR

f x j+1 | x j = f 1 x j+1 | x j + f 2 x j+1 | x j • 1 {0≤xj<xσ(xj)<xτ(xj)} + f 3 x j+1 | x j • 1 {xσ(xj)≤xj≤xτ(xj)} . (20)
In [START_REF] Chen | Condition-based maintenance using the inverse gaussian degradation model[END_REF], p 1 x j , p 2 x j , p 3 x j , p 4 x j and p 5 x j are respectively the transition probabilities conditional on X E + j = x j such that

• a PR or a CR starts after 1 inspection period (with X T j,1 ≥ x τ x j )

p 1 x j = F x τ x j -x j ; µ x j • δ, λ • δ 2 , (21)
• a PR or a CR starts after 1 inspection period (with x σ x j ≤ X T j,1 < x τ x j )

p 2 x j = x τ( x j) x σ( x j ) F x τ x j -w; µ x j • ψ α, w, x j , λ • ψ 2 α, w, x j • f w -x j ; µ x j • δ, λ • δ 2 dw, (22)
• a PR or a CR starts after at least 2 inspection periods (with X T j,k+1 ≥ x τ x j , k = 1, 2, . . .)

p 3 x j = x σ( x j ) x j F x τ x j -w; µ x j • δ, λ • δ 2 • ∞ k=1 f w -x j ; µ x j • kδ, λ • (kδ) 2 dw, (23)
• a PR or a CR starts after at least 2 inspection periods (with x σ x j ≤ X T j,k+1 < x τ x j , k = 1, 2, . . .)

p 4 x j = x τ (xj) x σ( x j) F x τ x j -z; µ x j • ψ α, z, x j , λ • ψ 2 α, z, x j ×        x σ( x j ) x j f z -w; µ x j • δ, λ • δ 2 • ∞ k=1 f w -x j ; µ x j • kδ, λ • (kδ) 2 dw        dz, (24)
• a PR or a CR starts without any inspection before

p 5 x j = F x τ x j -x j ; µ x j • ψ α, x j , x j , λ • ψ 2 α, x j , x j , (25)
while in [START_REF] Huynh | A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events[END_REF], f 1 x j+1 | x j , f 2 x j+1 | x j and f 3 x j+1 | x j are the transition pdf conditional on X E + j = x j such that

• an IR starts after 1 inspection period

f 1 x j+1 | x j = x τ( x j ) x σ( x j ) 1 {xj+1 0,x j ≤x j+1 ≤z} • g x j+1 | x j , z ×        x τ( x j) x σ (xj) 1 {w<z} • f z -w; µ x j • ψ α, w, x j , λ • ψ 2 α, w, x j • f w -x j ; µ x j • δ, λ • δ 2 dw        dz, (26)
• an IR starts after at least 2 inspection periods

f 2 x j+1 | x j = x τ( x j ) x σ( x j ) 1 {xj+1 0,x j ≤x j+1 ≤s} • g x j+1 | x j , s •        x τ (xj) x σ( x j ) 1 {z<s} • f s -z; µ x j • ψ α, z, x j , λ • ψ 2 α, z, x j ×        x σ( x j ) x j f z -w; µ x j • δ, λ • δ 2 • ∞ k=1 f w -x j ; µ x j • kδ, λ • (kδ) 2 dw        dz        ds, (27)
• an IR starts without any inspection before

f 3 x j+1 | x j =
x τ( x j )

x j

1 {xj+1 0,x j ≤x j+1 ≤w} • g x j+1 | x j , w • f w -x j ; µ x j • ψ α, x j , x j , λ • ψ 2 α, x j , x j dw, (28)
where f (•; •, •) and F (•; •, •) are derived from (3) and (5).

Stationary probability density function π dx j+1

From the closed-form expression (18) of P x j , dx j+1 , we can derive the stationary pdf π dx j+1 as a convex combination of Dirac mass function and a continuous pdf

π dx j+1 = a • δ 0 dx j+1 + (1 -a) • b x j+1 • 1 {0<xj+1<L} dx j+1 , (29)
where

a = 1 1 + L 0 B x j+1 dx j+1 and b x j+1 = a (1 -a) • B x j+1 . ( 30
)
The value of a has to belongs to [0, 1] as it is a probability. When a = 0, the system is never replaced at the end of a maintenance cycle; on the contrary, a replacement is always performed when a = 1. To obtain the values of B x j+1 , we divide the value interval (0, L) of x j+1 by N sub-intervals with same length h = L N : x j+1,0 = 0, x j+1,N = L, and x j+1,m = m • h with m = 1, . . . , N. Next, we approximate B x j+1 with 0 < x j+1 < L by B m = B x j+1,m with m = 1, . . . , N, such that

B = M -1 • q, (31)
where

B = B 0 • • • B N T , q = q 0 • • • q N T
, and M = M m,n 0≤m,n≤N is a lower triangular matrix with

M m,m = 1 - h 2 K m,m , m = 1, . . . , N, M m,n = -hK m,n , 1 ≤ n ≤ m ≤ N, M m,0 = - h 2 K m,0 , m = 1, . . . , N, M 0,0 = 1.
The quantities q m = q x j+1,m and K m,n = K x j+1,m | x j,n , with n = 1, . . . , m, are given by

q x j+1 = f 1 x j+1 | 0 + f 2 x j+1 | 0 • 1 {xj+1≤xτ(0)} • 1 {xσ(0)<xτ(0)} , (32)
and

K x j+1 | x j = f 1 x j+1 | x j + f 2 x j+1 | x j • 1 {0≤xj<xσ(xj)<xτ(xj)} + f 3 x j+1 | x j • 1 {xσ(xj)≤xj≤xτ(xj)} , (33)
where f 1 x j+1 | x j , f 2 x j+1 | x j and f 3 x j+1 | x j are given from ( 26), ( 27) and (28).

Numerical illustration

Based on the maintained system defined by L = 15, µ

X E + j = 1 + 0.1 • X E + j , λ = 4, ρ X E + j , X S j = ρ 0 + ρ 1 X E + j , X S j = 1 + 0.1 • X E + j + 0.2 • X S j
and a continuous uniform pdf for g x j+1 | X E + j , X S j , our goal is twofold: (i) show the shapes of P x j , dx j+1 and of π dx j+1 , and (ii) validate their mathematical development.

To illustrate both Dirac part and continuous part of P x j , dx j+1 , we consider two configurations: 0 ≤ x j < x σ x j < x τ x j :

x j = 3, δ = 3, σ = 1.1, α = 0.95, τ = 4, and x σ x j ≤ x j ≤ x τ x j : x j = 7, δ = 3, σ = 1.5, α = 0.95, τ = 4. The others (i.e., x τ x j ≤ x σ x j , 0 ≤ x j ≤ L and x σ x j < x τ x j ≤ x j < L ), which merely correspond to a Dirac measure at 0 with magnitude 1, are omitted. We next use two different methods to sketch out the shape of P x j , dx j+1 in Figures 4a and4b. The solid black curves are obtained by combining the Monte Carlo simulation (following the flowchart in Figure 1) and the kernel density estimation (KDE) [START_REF] Botev | Kernel density estimation via diffusion[END_REF]. Meanwhile, the dashed red curves are returned by the numerical computation of (18) using the well-known trapezoid rule. The identical results confirm the exactness of the mathematical formulation for P x j , dx j+1 .

Applying the above methods to the two following configurations of the (δ, σ, α, τ) policy:

Long-run maintenance cost rate evaluation and maintenance policy optimization

The aim of this section is to evaluate the long-run maintenance cost rate (13) using the semi-regenerative technique [START_REF] Grall | Continuous-time predictive-maintenance scheduling for a deteriorating system[END_REF]

C ∞ (δ, σ, α, τ) = lim t→∞ C (t) t = E π C E + j , E + j+1 E π ∆E j+1 , (34)
where C (t) is given from [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF], ∆E j+1 denotes the length of the (k + 1)-st Markov renewal cycle E + j , E + j+1 with X E + j = x j ∈ [0, L), E π [•] stands for the s-expectation with respect to the stationary measure π. Once the closed-form expression of C ∞ (δ, σ, α, τ) is available, we apply the generalized pattern search algorithm [START_REF] Audet | Derivative-Free and Blackbox Optimization[END_REF] to search the optimal configuration δ opt , σ opt , α opt , τ opt of the (δ, σ, α, τ) policy.

Long-run maintenance cost rate evaluation

From ( 12), we can express [START_REF] Huynh | Adaptive condition-based maintenance decision framework for deteriorating systems operating under variable environment and uncertain condition monitoring[END_REF] as

C ∞ (δ, σ, α, τ) = 1 E π ∆E j+1 • C m • E π N m E + j , E + j+1 + C r • E π N r E + j , E + j+1 +C o • E π N o E + j , E + j+1 + C i • E π I E + j , E + j+1 + C u • E π U E + j , E + j+1 , (35)
where

N m E + j , E + j+1 , N r E + j ,
E π N r E + j , E + j+1 + E π N o E + j , E + j+1 = 1. (36)
Moreover, we can express

E + j , E + j+1 as E + j , E + j + δ • N m E + j , E + j+1 ∪ E + j + δ • N m E + j , E + j+1 , S j ∪ S j , E + j+1 , so ∆E j+1 = δ • N m E + j , E + j+1 + W E + j , E + j+1 + I E + j , E + j+1 , (37)
where W E + j , E + j+1 denotes the waiting duration before a maintenance starts since the last inspection over E + j , E + j+1 . Consequently, the cost rate ( 35) is rewritten by

C ∞ (δ, σ, α, τ) = 1 δ • E π N m E + j , E + j+1 + E π W E + j , E + j+1 + E π I E + j , E + j+1 • C m • E π N m E + j , E + j+1 +C o -(C o -C r ) • E π N r E + j , E + j+1 + C i • E π I E + j , E + j+1 + C u • E π U E + j , E + j+1 . (38)
In the following, we give the mathematical expressions of

E π N m E + j , E + j+1 , E π N r E + j , E + j+1 , E π W E + j , E + j+1 , E π I E + j , E + j+1 and E π U E + j , E + j+1 .

Expected number of inspections over E

+ j , E + j+1 Starting at X E + j ∈ [0, L), the cycle E + j , E + j+1 may be stopped (i) after 1 inspection if 0 ≤ X E + j < x σ X E + j ≤ X T j,1 , (ii) after k + 1 with k ∈ N * if 0 ≤ X E + j < X T j,k < x σ X E + j ≤ X T j,k+1 , or even (iii) without inspection if x σ X E + j < X E + j < L.
So, the number of inspections over E + j , E + j+1 can be expressed as

N m E + j , E + j+1 = 1 0≤X E + j <x σ X E + j ≤X T j,1 + ∞ k=1 (k + 1) • 1 0≤X E + j <X T j,k <x σ X E + j ≤X T j,k+1 . ( 39
)
Its expected value with respect to the stationary measure π is thus computed by

E π N m E + j , E + j+1 = P m,1 + ∞ k=1 (k + 1) • P m,k+1 , (40)
where

• P m,1 denotes the probability that only one inspection is performed over

E + j , E + j+1 P m,1 = a • F x σ (0) ; µ (0) • δ, λ • δ 2 + (1 -a) • x σ( x j ) 0 F x σ x j -x j ; µ x j • δ, λ • δ 2 • b x j dx j , (41)
• P m,k+1 denotes the probability that (k + 1) inspections, with k ∈ N * , are performed over

E + j , E + j+1 P m,k+1 = a • x σ (0) 0 F x σ (0) -w; µ (0) • δ, λ • δ 2 • f w; µ (0) • kδ, λ • (kδ) 2 dw + (1 -a) × x σ( x j ) 0        x σ( x j) x j F x σ x j -w; µ x j • δ, λ • δ 2 • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw        • b x j dx j , ( 42
)
with a and b x j given from (30).

Expected number of repair over E

+ j , E + j+1
Since we can perform an IR over

E + j , E + j+1 (i) without inspection if x σ X E + j ≤ X E + j ≤ X S j < x τ X E + j , (ii) after 1 inspection if 0 ≤ X E + j < x σ X E + j ≤ X T j,1 ≤ X S j < x τ X E + j , or (iii) after (k + 1) inspections, with k ∈ N * , if 0 ≤ X E + j < X T j,k < x σ X E + j ≤ X T j,k+1 ≤ X S j < x τ X E + j , the number of IR over E + j , E + j+1 can be expressed by N r E + j , E + j+1 = 1 x σ X E + j ≤X E + j ≤X S j <x τ X E + j + 1 0≤X E + j <x σ X E + j ≤X T j,1 ≤X S j <x τ X E + j + ∞ k=1 1 0≤X E + j <X T j,k <x σ X E + j ≤X T j,k+1 ≤X S j <x τ X E + j . ( 43
)
This leads us to compute

E π N r E + j , E + j+1 as E π N r E + j , E + j+1 = P r,0 + P r,1 + ∞ k=1 P r,k+1 , (44)
where

• P r,0 denotes the probability that an IR is performed over E + j , E + j+1 without any inspection

P r,0 = (1 -a) • x τ (xj) x σ( x j ) F x τ x j -x j ; µ x j • ψ α, x j , x j , λ • ψ 2 α, x j , x j • b x j dx j , (45)
• P r,1 denotes the probability that an IR is performed over E + j , E + j+1 after one inspection

P r,1 = a • x τ (0) x σ (0) F x τ (0) -w; µ (0) • ψ (α, w, 0) , λ • ψ 2 (α, w, 0) • f w; µ (0) • δ, λ • δ 2 dw + (1 -a) • x σ( x j) 0 b x j ×        x τ (xj) x σ( x j) F x τ x j -w; µ x j • ψ α, w, x j , λ • ψ 2 α, w, x j • f w -x j ; µ x j • δ, λ • δ 2 dw        dx j , (46)
• P r,k+1 denotes the probability that an IR is performed over

E + j , E + j+1 after (k + 1) inspections with k ∈ N * P r,k+1 = a • x σ (0) 0 x τ (0) x σ (0) F x τ (0) -z; µ (0) • ψ (α, z, 0) , λ • ψ 2 (α, z, 0) × f z -w; µ (0) • δ, λ • δ 2 dz • f w; µ (0) • kδ, λ • (kδ) 2 dw + (1 -a) • x σ( x j ) 0        x σ( x j) x j        x τ (xj) x σ( x j ) F x τ x j -z; µ x j • ψ α, z, x j , λ • ψ 2 α, z, x j × f z -w; µ x j • δ, λ • δ 2 dz • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw • b x j dx j , ( 47
)
with a and b x j given from (30).

Expected length of waiting duration over E

+ j , E + j+1
Following the (δ, σ, α, τ) policy, the considered waiting duration is the time interval from the latest inspection (including the starting time of a Markov renewal cycle) to the beginning of a maintenance action. Since a maintenance can be done without inspection, after one inspection or after (k + 1) inspections with k ∈ N * , the associated waiting duration over

E + j , E + j+1 is either (i) ψ α, X E + j , X E + j if x σ X E + j ≤ X E + j < L, (ii) ψ α, X T j,1 , X E + j if 0 ≤ X E + j < x σ X E + j ≤ X T j,1 < L, or (iii) ψ α, X T j,k+1 , X E + j if 0 ≤ X E + j < X T j,k < x σ X E + j ≤ X T j,k+1 < L.
We can therefore express the waiting duration W E + j , E + j+1 by

W E + j , E + j+1 = ψ α, X E + j , X E + j • 1 x σ X E + j ≤X E + j <LL + ψ α, X T j,1 , X E + j • 1 0≤X E + j <x σ X E + j ≤X T j,1 <L + ∞ k=1 ψ α, X T j,k+1 , X E + j • 1 0≤X E + j <X T j,k <x σ X E + j ≤X T j,k+1 <L . ( 48
)
As such, we obtain its expected value with respect to π as

E π W E + j , E + j+1 = E w,0 + E w,1 + ∞ k=1 E w,k+1 , (49)
where

• E w,0 denotes the expected length of the waiting time just after the beginning of the renewal cycle (i.e., without inspection)

E w,0 = (1 -a) • L x σ( x j ) ψ α, x j , x j • b x j dx j . (50)
• E w,1 denotes the expected length of the waiting time just the 1-st inspection

E w,1 = a • L x σ (0) ψ (α, w, 0) • f w; µ (0) • δ, λ • δ 2 dw + (1 -a) • x σ( x j) 0       L x σ( x j ) ψ α, w, x j • f w -x j ; µ x j • δ, λ • δ 2 dw       • b x j dx j , (51)
• E w,k+1 denotes the expected length of the waiting time just the (k + 1)-st inspection

E w,k+1 = a • x σ (0) 0 L x σ (0) ψ (α, z, 0) • f z -w; µ (0) • δ, λ • δ 2 dz • f w; µ (0) • kδ, λ • (kδ) 2 dw + (1 -a) • x σ( x j ) 0        x σ( x j) x j       L x σ (xj) ψ α, z, x j f z -w; µ x j • δ, λ • δ 2 dz       × f w -x j ; µ x j • kδ, λ • (kδ) 2 dw • b x j dx j , (52)
with a and b x j given from (30).

Expected length of the system inactivity duration over E

+ j , E + j+1
During a preventive maintenance (i.e., either a PR or an IR), the considered system is inactivated. Such an inactivity duration over E + j , E + j+1 can be obtained by analyzing all possible scenarios of PR or IR

I E + j , E + j+1 = ρ 0 • 1 all scenarios of PR and IR over E + j ,E + j+1 + ρ 1 X E + j , X S 0 • 1 all scenarios of IR over E + j ,E + j+1 . (53)
However, since the scenarios of PR are burdensome to analyze, we propose to base on the ones of CR or IR instead

I E + j , E + j+1 = ρ 0 • 1 -1 all scenarios of CR over E + j ,E + j+1 + ρ 1 X E + j , X S 0 • 1 all scenarios of IR over E + j ,E + j+1 . (54)
All scenarios for IR have been analyzed in Subsection 4.1.2, it is enough to seek the scenarios of CR to make clear [START_REF] ¸inlar | Periodicity in markov renewal theory[END_REF]. Over

E + j , E + j+1 , a CR may be carried out either 1. without inspection if x σ X E + j ≤ X E + j < L ≤ X S 0 , 2. with one inspection: (i) at the 1-st inspection if 0 ≤ X E + j < x σ X E + j < L ≤ X T j,1 , (ii) after the 1-st inspection if 0 ≤ X E + j < x σ X E + j < X T j,1 < L ≤ X S 0 , 3. or with k + 1 inspections, k ∈ N * : (i) at the (k + 1)-st inspection if 0 ≤ X E + j < X T j,k < x σ X E + j < L ≤ X T j,k+1 , (ii) after the (k + 1)-st inspection if 0 ≤ X E + j < X T j,k < x σ X E + j < X T j,k+1 < L ≤ X S 0 .
Therefore, we can express I E + j , E + j+1 as

I E + j , E + j+1 = ρ 0 1 -1 x σ X E + j ≤X E + j <L≤X S 0 -1 0≤X E + j <x σ X E + j <L≤X T j,1 -1 0≤X E + j <x σ X E + j <X T j,1 <L≤X S 0 - ∞ k=1 1 0≤X E + j <X T j,k <x σ X E + j <L≤X T j,k+1 - ∞ k=1 1 0≤X E + j <X T j,k <x σ X E + j <X T j,k+1 <L≤X S 0        + ρ 1 X E + j , X S 0 • 1 0≤x σ X E + j ≤X E + j ≤X S 0 <x τ X E + j + ρ 1 X E + j , X S 0 • 1 0≤X E + j <x σ X E + j ≤X T j,1 ≤X S 0 <x τ X E + j + ∞ k=1 ρ 1 X E + j , X S 0 • 1 0≤X E + j <X T j,k <x σ X E + j ≤X T j,k+1 ≤X S 0 <x τ X E + j . (55)
Its expectation with respect to π is computed by

E π I E + j , E + j+1 = ρ 0 •        1 -P c,0 -P c,1,i -P c,1,s - ∞ k=1 P c,k+1,i - ∞ k=1 P c,k+1,s        + E i,r,0 + E i,r,1 + ∞ k=1 E i,r,k+1 , (56)
where

• P c,0 denotes the probability that a CR is performed without any inspection

P c,0 = (1 -a) • L x σ( x j ) F L -x j ; µ x j • ψ α, x j , x j , λ • ψ 2 α, x j , x j • b x j dx j , (57)
• P c,1,i denotes the probability that a CR is performed at the 1-st inspection

P c,1,i = a • F L; µ (0) • δ, λ • δ 2 + (1 -a) • x σ( x j ) 0 F L -x j ; µ x j • δ, λ • δ 2 • b x j dx j , (58)
• P c,1,s denotes the probability that a CR is performed after the 1-st inspection

P c,1,s = a • L x σ (0) F L -w; µ (0) • ψ (α, w, 0) , λ • ψ 2 (α, w, 0) • f w; µ (0) • δ, λ • δ 2 dw + (1 -a) • x σ( x j ) 0 b x j ×       L x σ( x j) F L -w; µ x j • ψ α, w, x j , λ • ψ 2 α, w, x j • f w -x j ; µ x j • δ, λ • δ 2 dw       dx j , (59)
• P c,k+1,i denotes the probability that a CR is performed at the (k + 1)-st inspection

P c,k+1,i = a • x σ (0) 0 F L -w; µ (0) • δ, λ • δ 2 • f w; µ (0) • kδ, λ • (kδ) 2 dw + (1 -a) × x σ( x j ) 0        x σ( x j) x j F L -w; µ x j • δ, λ • δ 2 • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw        • b x j dx j , (60)
• P c,k+1,s denotes the probability that a CR is performed after the (k + 1)-st inspection

P c,k+1,s = a • x σ (0) 0 L x σ (0) F L -z; µ (0) • ψ (α, z, 0) , λ • ψ 2 (α, z, 0) • f z -w; µ (0) • δ, λ • δ 2 dz × f w; µ (0) • kδ, λ • (kδ) 2 dw + (1 -a) • x σ( x j ) 0        x σ( x j) x j       L x σ (xj) F L -z; µ x j • ψ α, zx j , λ • ψ 2 α, z, x j × f z -w; µ x j • δ, λ • δ 2 dz • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw • b x j dx j , (61)
• E i,r,0 denotes the expected length of the system inactivity when no inspection is performed over

E + j , E + j+1 E i,r,0 = (1 -a) • x τ (xj) x σ( x j) b x j        x τ (xj) x j ρ 1 x j , w • f w -x j ; µ x j • ψ α, x j , x j , λ • ψ 2 α, x j , x j dw        dx j , (62)
• E i,r,1 denotes the expected length of the system inactivity when 1 inspection is performed over

E + j , E + j+1 E i,r,1 = a • x τ (0)
x σ (0)

x τ (0) w ρ 1 (0, z) • f z -w; µ (0) • ψ (α, w, 0) , λ • ψ 2 (α, w, 0) dz • f w; µ (0) • δ, λ • δ 2 dw + (1 -a) • x σ( x j) 0        x τ (xj) x σ( x j )        x τ (xj) w ρ 1 x j , z • f z -w; µ x j • ψ α, w, x j , λ • ψ 2 α, w, x j dz        × f w -x j ; µ x j • δ, λ • δ 2 dw • b x j dx j , (63)
• E i,r,k+1 denotes the expected length of the system inactivity when (k + 1) inspections are performed over

E + j , E + j+1 E i,r,k+1 = a • x σ (0) 0 x τ (0)
x σ (0)

x τ (0) z ρ 1 (0, s) • f s -z; µ (0) • ψ (α, z, 0) , λ • ψ 2 (α, z, 0) ds × f z -w; µ (0) • δ, λ • δ 2 dv • f w; µ (0) • kδ, λ • (kδ) 2 dz + (1 -a) • x σ( x j ) 0        x σ( x j ) x j        x τ (xj) x σ( x j)        x τ( x j ) z ρ 1 x j , s • f s -z; µ x j • ψ α, z, x j , λ • ψ 2 α, z, x j ds        × f z -w; µ x j • δ, λ • δ 2 dv • f w -x j ; µ x j • kδ, λ • (kδ) 2 dz • b x j dx j , (64)
with a and b x j given from (30).

Expected length of the system unavailability duration over E

+ j , E + j+1
Once the system fails, it is unavailable from the failure time to starting time of the next CR and then during the CR duration. This is why an analysis of scenarios of system failure and CR allows to determine the unavailability duration of the system. In fact, the duration of system unavailability over

E + j , E + j+1 is either (i) ψ α,X E + j ,X E + j 0 1 x σ X E + j ≤X E + j <L≤X t dt + ρ 0 • 1 x σ X E + j ≤X E + j <L≤X S 0
if the system fails before the first inspection and a CR is performed without inspection, (ii)

δ 0 1 0≤X E + j <x σ X E + j <L≤X t dt + ρ 0 • 1 0≤X E + j <x σ X E + j <L≤X T j,1
if the system failure is detected at the 1-st inspection at which a CR starts, (iii)

δ+ψ α,X T j,1 ,X E + j δ 1 0≤X E + j <x σ X E + j <X T j,1 <L≤X t dt + ρ 0 • 1 0≤X E + j <x σ X E + j <X T j,1 <L≤X S 0
if the system failure is detected after the 1-st inspection at which a CR starts, (iv)

(k+1)δ kδ 1 0≤X E + j <X T j,k <x σ X E + j <L≤X t dt + ρ 0 • 1 0≤X E + j <X T j,k <x σ X E + j <L≤X T j,k+1
if the system failure is detected at the (k + 1)-st inspection with k ∈ N * at which a CR starts,

(v) (k+1)δ+ψ α,X T j,k+1 ,X E + j (k+1)δ 1 0≤X E + j <X T j,k <x σ X E + j <X T j,k+1 <L≤X t dt+ρ 0 •1 0≤X E + j <X T j,k <x σ X E + j <X T j,k+1 <L≤X S 0
if the system failure is detected after the (k + 1)-st inspection with k ∈ N * at which a CR starts.

As a result, we can express the expected length of the system unavailability duration over E + j , E + j+1 with respect to the stationary measure π as

E π U E + j , E + j+1 = ρ 0 •        P c,0 + P c,1,i + P c,1,s + ∞ k=1 P c,k+1,i + ∞ k=1 P c,k+1,s        + E u,0 + E u,1,i + E u,1,s + ∞ k=1 E u,k+1,i + ∞ k=1 E u,k+1,s , (65)
where P c,0 , P c,1,i , P c,1,s , P c,k+1,i and P c,k+1,s are given from ( 57), ( 58), ( 59), ( 60) and ( 61) respectively, and E u,0 , E u,1,i , E u,1,s , E u,k+1,i and E u,k+1,s are represented as follows.

• E u,0 denotes the expectation of the duration from the system failure time to the starting time of the next CR when no inspection has been performed

E u,0 = (1 -a) • L x σ( x j )        ψ(α,x j ,x j) 0 F L -x j ; µ x j • t, λ • t 2 dt        • b x j dx j , (66)
• E u,1,i denotes the expectation of the duration from the system failure time to the 1-st inspection time at which the CR starts

E u,1,i = a • δ 0 F L; µ (0) • t, λ • t 2 dt + (1 -a) • δ 0        x σ( x j ) 0 F L -x j ; µ x j • t, λ • t 2 • b x j dx j        dt, (67)
• E u,1,s denotes the expectation of the duration from the system failure time to the starting time of the next CR when one inspection has been done

E u,1,s = a • L x σ (0) ψ(α,w,0) 0 F L -w; µ (0) • t, λ • t 2 dt • f w; µ (0) • δ, λ • δ 2 dw + (1 -a) • x σ( x j ) 0 b x j •        L x σ( x j)        ψ(α,w,x j) 0 F L -w; µ x j • θ, λ • θ 2 dθ        • f w -x j ; µ x j • δ, λ • δ 2 dw        dx j , (68)
• E u,k+1,i denotes the expectation of the duration from the system failure time to the (k + 1)-st inspection time at which the CR starts

E u,k+1,i = a • δ 0 x σ (0) 0 F L -w; µ (0) • t, λ • t 2 • f w; µ (0) • kδ, λ • (kδ) 2 dw dt + (1 -a) • δ 0        x σ( x j) 0        x σ (xj) x j F L -w; µ x j • t, λ • t 2 • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw        • b x j dx j        dt, (69)
• E u,k+1,s denotes the expectation of the duration from the system failure time to the starting time of the next CR when (k + 1)

inspections have been done

E u,k+1,s = a • x σ (0) 0 L x σ (0) ψ(α,z,0) 0 F L -z; µ (0) • t, λ • t 2 dt • f z -w; µ (0) • δ, λ • δ 2 dz × f w; µ (à) • kδ, λ • (kδ) 2 dw + (1 -a) • x σ( x j ) 0        x σ( x j) x j        L x σ( x j )        ψ(α,z,x j) 0 F L -z; µ x j • t, λ • t 2 dt        × f z -w; µ x j • δ, λ • δ 2 dz • f w -x j ; µ x j • kδ, λ • (kδ) 2 dw • b x j dx j , (70)
with a and b x j given from (30).

Maintenance policy optimization

Optimizing the (δ, σ, α, τ) policy returns to seek the optimal configuration δ opt , σ opt , α opt , τ opt that minimizes the long-run maintenance cost rate C ∞ (δ, σ, α, τ)

δ opt , σ opt , α opt , τ opt = arg min (δ,σ,α,τ) {C ∞ (δ, σ, α, τ) : δ > 0, σ > 0, 0 < α < 1, τ ≥ ρ 0 } , (71)
where ρ 0 denotes the required duration for a replacement. Even though the closed-form expression of C ∞ (δ, σ, α, τ) is available, its complexity does not allow an analytical solution for (71), and numerical methods should be used instead.

To prove the existence of δ opt , σ opt , α opt , τ opt numerically, we vary δ, σ, α and τ in a wide rank, and observe the form of C ∞ (δ, σ, α, τ). Repeating such an experiment for various configurations of maintenance costs (i.e., C m , C r , C o , C i and C u ) and of system characteristics (i.e., L, µ 0 , µ 1 (•), λ, ρ 0 , ρ 1 (•, •), g (• | •, •)), we can draw a conclusion about the existence of δ opt , σ opt , α opt , τ opt . Although this numerical approach cannot cover all possible configurations of the maintained system, it is still an alternative solution when analytical approach is impossible. To find the optimum δ opt , σ opt , α opt , τ opt , we propose using the generalized pattern search algorithm [START_REF] Audet | Derivative-Free and Blackbox Optimization[END_REF]. Numerous numerical experiments confirm that this algorithm allows to speed up the optimum finding regardless of chosen initial values.

Numerical illustration

As an illustration, we apply the maintenance costs C m = 2, C r = 20, C o = 100, C i = 5 and C u = 15 to the system considered in Subsection 3.3. We compute and sketch in Figure 6 the long-run maintenance cost rate C ∞ (δ, σ, α, τ) by varying δ from 1 to 6 with step 0.25, σ from 0.6 to 1.6 with step 0.05, α from 0.79 to 0.99 with step 0.01, and τ from 3 to 5 with step 0.1. In the Figures 6a, 6b confirm the existence of optimal configuration for the (δ, σ, α, τ) policy.

In Figure 6, we find the minimum value C ∞ (δ * , σ * , α * , τ * ) = 6.8163 at δ * = 4.1, σ * = 1.2, α * = 0.94, τ * = 4.1. However, this configuration is not optimal yet, because the chosen lattices for δ, σ, α and τ are not fine enough. To seek the "real" optimum δ opt , σ opt , α opt , τ opt , we use the patternsearch solver of Matlab's Global Optimization Toolbox. As shown in Figure 7, the optimal configuration of the above maintained system is reached at δ opt = 3.375, σ opt = 1.1563, α opt = 0.94688 and τ opt = 4 with C ∞ δ opt , σ opt , α opt , τ opt = 6.8085.

Numerical assessment

To assess the economic performances of the (δ, σ, α, τ) policy, comparative studies with benchmark policies are proposed in this section. The studies take into account the impacts of different maintenance costs and various system deterioration characteristics.

Notwithstanding, we just present hereinafter the results for the maintained system defined by • a degradation process with linear shape function:

IGP µ x j , λ = IGP µ 0 + µ 1 • x j , λ = IGP 1 + µ 1 • x j , 4 , • a linear IR duration: ρ X E + j , X S j = ρ 0 + ρ 1 X E + j , X S j = ρ 0 + ρ 1,1 • X E + j + ρ 1,2 • X S j = 1 + 0.1 • X E + j + 0.2 • X S j ,
• a continuous uniform pdf for g x j+1 | X E + j , X S j .

The linear form of µ x j and ρ 1 X E + j , X S j is inspired by [START_REF] Guo | Improved inverse gaussian process and bootstrap: Degradation and reliability metrics[END_REF] and [START_REF] Bérenguer | Maintenance policy for a continuously monitored deteriorating system[END_REF] respectively, while the other parameters are arbitrarily chosen. The applied maintenance costs are fixed at C m = 2, C o = 100, C i = 5 and C u = 15. Some missing values (i.e., µ 1 and C r ) will be stated latter depending on concrete numerical illustrations.

Benchmark maintenance policies

Three PdM policies are used as benchmarks. The first one, called (δ, ζ, ω, η), is static in the sense that repair and replacement decisions are not adaptive to the system degradation behavior. Meanwhile, as extreme cases of the (δ, σ, α, τ) policy, the two others, called (δ, σ, α, τ → ρ 0 ) and (δ, σ, α, τ → +∞), are adaptive, but either the IR or perfect replacement is used as preventive actions.

(δ, ζ, ω, η) policy

The (δ, ζ, ω, η) policy generalizes the PdM policies proposed in [START_REF] Huynh | Modeling past-dependent partial repairs for condition-based maintenance of continuously deteriorating systems[END_REF] and [START_REF] Huynh | A parametric predictive maintenance decision-making framework considering improved system health prognosis precision[END_REF]. Over the cycle E + j , E + j+1 , the system is regularly inspected at times T j,k = E j + k • δ, with k = 0, 1, 2, . . .. to reveal its degradation level. Given X T j,k , we adopt the following decision rules.

1. If X T j,k ≥ L, the failed system is correctively replaced immediately at T j,k . After the CR with duration ρ 0 , the system is AGAN (i.e., X E + j+1 = 0 with E j+1 = T j,k + ρ 0 ). 2. If ξ ≤ X T j,k < L, the system is still running at T j,k and its RUL can be predicted with an acceptable precision (see Subsection 2.2). So, no further inspection is needed, and the next maintenance is planned ω ≥ 0 time units later (i.e., at S j = T j,k + ω). The nature of the maintenance action depends on both the degradation levels X S j and X E + j . (a) If X S j ≥ L, a CR with duration ρ 0 is triggered immediately at S j to reset the failed system to an AGAN state (i.e., X E + j+1 = 0 with E j+1 = S j + ρ 0 ). (b) If X S j < L and X E + j ≥ η, the system is still running at S j , and the "high" value of X E + j implies that an IR is no longer suitable for the current maintenance. So, a PR should be carried out at S j instead. After the PR, the system is AGAN (i.e., X E + j+1 = 0 with E j+1 = S j + ρ 0 ). (c) If X S j < L and X E + j < η, a preventive IR is immediately performed at S j on the running system. It takes ρ X E + j , X S j time units, and returns the system degradation X E + j+1 to a random level between X E + j and X S j such that X E + j+1 ∼ g y | X E + j , X S j , where E j+1 = S j + ρ X E + j , X S j and g is a known truncated pdf. 3. If X T j,k < ξ, additional inspections are required to reinforce the precision of RUL prediction. Accordingly, the decisions is postponed to the next inspection time at T j,k+1 = T j,k + δ. The next maintenance cycle begins at E + j+1 with initial deterioration level X E + j+1 . The inspection period δ, the degradation thresholds ξ and η, and the waiting time ω are decision variables. Its long-run maintenance cost model is developed in the same way as the (δ, σ, α, τ) policy. Figure 8 illustrates the schematic behavior of the maintained system under the (δ, ζ, ω, η) policy. Its meaning is very similar to the one of Figure 3. Clearly, with fixed ξ, ω and η, the maintenance decisions of the (δ, ζ, ω, η) policy cannot adapt to the system degradation behavior. So, the comparison between the (δ, ζ, ω, η) and (δ, σ, α, τ) policies allows to see the added values of adaptive maintenance decisions.

5.1.2.

(δ, σ, α, τ → ρ 0 ) policy and (δ, σ, α, τ → +∞) policy These two benchmarks policies are extreme cases of the (δ, σ, α, τ) policy, where either replacement or repair is used as preventive maintenance action.

1. When τ → ρ 0 , only replacement is implemented for preventive action because P ρ X E + j , X S j ≥ τ → 1. The (δ, σ, α, τ) policy becomes a pure PR policy (δ, σ, α, τ → ρ 0 ) (see also [START_REF] Huynh | A parametric predictive maintenance decision-making framework considering improved system health prognosis precision[END_REF]). The schematic behavior of the maintained system under the (δ, σ, α, τ → ρ 0 ) policy is sketched in Figure 9. We find that the waiting time ψ (•) well adapts to the system degradation. 2. When τ → +∞, only repair is carried out in preventive decision because P ρ X E + j , X S j ≥ τ → 0. So, (δ, σ, α, τ) policy returns to a pure preventive IR policy (δ, σ, α, τ → +∞). The schematic behavior of the associated maintained system is illustrated in Figure 10. Obviously, the flexibility of the (δ, σ, α, τ → +∞) policy is reflected not only by the dynamic waiting time ψ (•), but also by the varied value of x σ .

The comparison between the (δ, σ, α, τ) policy and its extreme cases will justify the effectiveness of hybrid maintenance decisions.

Case studies and comparison results

To understand the impacts of the maintenance costs and the system characteristic on the cost saving, we consider 2 following case studies:

1. sensitivity to repair cost: C r varies from 3 to 39 with step 3, and µ 1 = 0.1, 2. sensitivity to degradation rate: C r = 20, and µ 1 varies from 0 to 0.3 with step 0.03.

The associated optimal long-run maintenance cost rate of the 4 considered maintenance policies are shown in Figures 11a and11b.

In each figure, lowers curve correspond to higher economic performances. The (δ, σ, α, τ → ρ 0 ) policy has a constant optimal cost rate in both the situations, because this pure PR policy is independent of C r and λ 1 . Meanwhile, using IR as a preventive maintenance action, the other PdM policies have evidently increasing cost rate with respect to C r and λ 1 . Comparing the cost curves of the (δ, σ, α, τ → ρ 0 ) policy and (δ, σ, α, τ → +∞) policy, we find out the effectiveness of each kind of maintenance actions. The replacement is better if C r or µ 1 is small, otherwise the repair is more profitable.

To make use of their advantage, hybrid maintenance decisions should be resorted to. Indeed, as clearly shown in Figures 11a and11b, the (δ, σ, α, τ) policy always saves more maintenance cost, and just returns to the pure policies in worse cases. Now, looking at the cost curves of the (δ, σ, α, τ) policy and the (δ, ζ, ω, η) policy, the former always gives more cost saving. The profit is even higher when C r or µ 1 increases. This implies that adaptive decisions allows to resist better to the negative impact of maintenance costs, and to well adapt to the variation of the system degradation behavior.

Conclusion and perspectives

The focus of this paper is to develop a cost-effective PdM model for deteriorating systems using periodic inspection, IR and perfect replacement. The development consists of four steps: continuous degradation modeling, maintenance effect modeling, adaptive PdM policy elaboration, and long-run maintenance cost rate evaluation. The connection between these steps is especially highlighted by (i) the consideration of the past dependency of IR actions in the IG degradation process, (ii) the use of estimated system RUL and maintenance duration to enable adaptive PdM decisions, and (iii) the probabilistic study of the behavior of maintained system at steady state based on the semi-regenerative theory. Various numerical experiments and comparative studies show that the developed adaptive PdM model is more flexible, and hence more profitable than related PdM models.

Despite very encouraging results, our PdM model is still based on a strong assumption that the model parameters are already known. However, these parameters are usually unknown in practice, and should be estimated from the available degradation and maintenance data. This is why our future works will focus on overcoming this drawback. Some potential perspectives are as follows. Firstly, we think of building a testing platform able to deliver both the degradation and maintenance data. The reason is that most existing benchmark data-sets1 are interesting to test prognostic algorithms, but not suitable for the PdM modeling. Once the required data are available, we carry out the distribution selection for the past-dependent IR and the parameters estimation for the IG degradation process. We believe that statistical methods proposed in [START_REF] Guo | Improved inverse gaussian process and bootstrap: Degradation and reliability metrics[END_REF][START_REF] Liu | A data-driven approach to selecting imperfect maintenance models[END_REF][START_REF] Salles | Semiparametric estimate of the efficiency of imperfect maintenance actions for a gamma deteriorating system[END_REF] can help. Finally, we shall adapt our PdM model to online application by updating the decision variables following available monitoring data. Such an online model is currently studied in [START_REF] Omshi | A dynamic auto-adaptive predictive maintenance policy for degradation with unknown parameters[END_REF] under Bayesian framework, but the considered PdM policy remains relatively simple.

Figure 1 :

 1 Figure 1: Maintenance plan and decision rules over the ( j + 1)-st maintenance cycle E + j , E + j+1

  FR T j,k |X T j,k ,X E + j r | x j,k , x j

  b) RUL sdt at inspection times

Figure 3 :

 3 Figure 3: Schematic behavior of the maintained system under the policy (δ, σ, α, τ)

  j j∈N, where Y j = X E + j denotes the system degradation level at the semiregenerative time E + j . The Markov chain Y j j∈N starts from Y 0 = 0, takes the value in the continuous state space [0, L), and comes back to 0 (i.e., a regeneration set) almost surely due to replacement actions. According to [55, Chapter VII, Section 3], Y j j∈N is Harris recurrent and has an unique stationary measure π which is the solution of the following invariance equation[START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] Chapter 10]

4 Figure 4 :

 44 Figure 4: Shapes of the transition pdf P x j , dx j+1

= 3 ,( 1

 31 σ = 1.1, α = 0.95, τ = 4, and δ = 3, σ = 1.5, α = 0.95, τ = 4, we obtain in Figures 5a and 5b the shapes of π dx j+1 . Once again, the identical results given by a) • b(x j+1 ) • 1 {0<xj+1<L} • dx j+1 M.C. Sim. Num. Comp.

(a) δ = 3 ,

 3 σ = 1.1, α = 0.95 and τ = 4

( 1 4 Figure 5 :

 145 Figure 5: Shapes of the stationary pdf π dx j+1

Figure 6 :

 6 Figure 6: Shapes of C ∞ (δ, σ, α, τ) and C ∞ (δ * , σ * , α * , τ * ) = 6.8163 and 6c, 2 among 4 decision variables are fixed and 2 others vary, while in Figure 6d, only δ varies. The convex forms of C ∞ (δ, σ, α, τ)

Figure 7 :

 7 Figure 7: Optimization of (δ, σ, α, τ) policy with Matlab's patternsearch solver

Figure 8 :

 8 Figure 8: Schematic behavior of the maintained system with δ = 2.33, ξ = 11.03, ω = 1 and η = 3.53 (i.e., optimal (δ, ζ, ω, η) policy when µ 1 = 0.1 and C r = 20)

ψFigure 9 :

 9 Figure9: Schematic behavior of the maintained system with δ = 5.75, σ = 1.3, α = 0.939 (i.e., optimal (δ, σ, α, τ → ρ 0 ) policy when µ 1 = 0.1 and C r= 20)

Figure 10 :

 10 Figure 10: Schematic behavior of the maintained system with δ = 5.75, σ = 1.3, α = 0.939 (i.e., optimal (δ, σ, α, τ → +∞) policy when µ 1 = 0.1 and C r = 20)

1 Figure 11 :

 111 Figure 11: Evolution of the optimal long-run maintenance cost rates

See e.g., NASA prognostic data repository: https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/, or CALCE Battery Research Group: https://web.calce.umd.edu/batteries/data.htm

Acknowledgment

This work was partially supported by the Fond Européen de Développement Régional (FEDER) under the framework of Project MAP 4.0.