K T Huynh
email: tuan.huynh@utt.fr

Modeling past-dependent partial repairs for condition-based maintenance of continuously deteriorating systems

Keywords: Maintenance, past dependency, partial repair, deterioration process, semi-regenerative process

We are interested in the stochastic modeling of a condition-based maintained system subject to continuous deterioration and maintenance actions such as inspection, partial repair and replacement. The partial repair is assumed dependent on the past in the sense that it cannot bring the system back into a deterioration state better than the one reached at the last repair. Such a past-dependency can affect (i) the selection of a type of maintenance actions, (ii) the maintenance duration, (iii) the deterioration level after a maintenance, and (iv) the restarting system deterioration behavior. In this paper, all these effects are jointly considered in an unifying condition-based maintenance model on the basis of restarting deterioration states randomly sampled from a probability distribution truncated by the deterioration levels just before a current repair and just after the last repair/replacement. Using results from the semi-regenerative theory, the long-run maintenance cost rate is analytically derived. Numerous sensitivity studies illustrate the impacts of past-dependent partial repairs on the economic performance of the considered condition-based maintained system.

Acronyms

Introduction

Maintenance is an effective solution to reduce the system failure, improve the system availability, and extend the system lifetime. It has been adopted in a wide range of systems, such as civil infrastructure [START_REF] Frangopol | Maintenance and management of civil infrastructure based on condition, safety, optimization, and life-cycle cost[END_REF], manufacturing systems [START_REF] Lee | Self-maintenance and engineering immune systems: Towards smarter machines and manufacturing systems[END_REF], automotive vehicles [START_REF] Lu | A review of innovations in disbonding techniques for repair and recycling of automotive vehicles[END_REF], I.T. software [START_REF] Benestad | Understanding software maintenance and evolution by analyzing individual changes: a literature review[END_REF], energy assets [START_REF] Shafiee | Maintenance optimization and inspection planning of wind energy assets: Models, methods and strategies[END_REF], etc. Maintenance activities comprise perfect and imperfect actions classified following their effects on the condition of maintained systems [START_REF] Pham | Imperfect maintenance[END_REF]. If the maintenance recovers the system back to an as-good-as new (AGAN) condition, it is perfect; otherwise, it is imperfect. Typical examples of perfect maintenance are complete replacement and overhaul, and of imperfect maintenance are testing, inspection, minimal repair and partial repair. Unlike restricted applications of perfect maintenance in engineering practice, imperfect maintenance characterizes divers realistic actions whose imperfectness may be caused by various factors such as human errors, spare parts quality, lack of materials, lack of maintenance time, etc. Modeling imperfect maintenance is thus crucial for practical needs.

The present paper deals with a particular kind of imperfect maintenance called past-dependent partial repair. It is characterized by the phenomenon that a partial repair cannot bring a deteriorating system back into a deterioration state better than the one reached at the last repair. The deterioration paths of draught fans and of gyroscopes provided in (Wang et al., 2018) and [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF][START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF] are some real-world examples for this phenomenon. Our aim is to develop a condition-based maintenance (CBM) with past-dependent partial repairs for continuously deteriorating systems considering. The main motivation is that CBM is usually more efficient than run-to-failure maintenance and timebased maintenance (TBM) [START_REF] Ahmad | An overview of time-based and condition-based maintenance in industrial application[END_REF], especially when the system deterioration is nowadays easily accessed thanks to the development of sensor and data transmission technologies [START_REF] Roy | Continuous maintenance and the future-foundations and technological challenges[END_REF]. Notwithstanding, the literature of past-dependent partial repairs is mostly attached to TBM via models with memory (e.g., arithmetic reduction of age (ARA) and arithmetic reduction of intensity [START_REF] Doyen | Classes of imperfect repair models based on reduction of failure intensity or virtual age[END_REF]). Therefore, modeling this kind of imperfect maintenance is still a widely open issue in the contexte of CBM.

In the literature, three main approaches have been employed to take into account the past dependency in imperfect CBM models applied to continuously deteriorating systems.

1. Repairs number-based modeling. The first approach considers that the system residual damage after each partial repair exhibits an increasing trend with the sequence of repairs. Since the number of repairs increases over time until the next replacement, their ability to improve the system deterioration weakens. This leads to the dependency between past and current repairs. Through the repairs number, past dependency effects on the maintained system are modeled. For instance, [START_REF] Liao | Maintenance of continuously monitored degrading systems[END_REF]; [START_REF] Guo | A maintenance optimization model for mission-oriented systems based on wiener degradation[END_REF]; [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF]; [START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF] have applied this approach to express the past dependency of both the system deterioration level and deterioration rate. Besides, in [START_REF] Liao | Maintenance of continuously monitored degrading systems[END_REF][START_REF] Guo | A maintenance optimization model for mission-oriented systems based on wiener degradation[END_REF], a higher number of performed repairs also lengthen maintenance duration, but it does not contribute to CBM decision-making. On the contrary, in [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF][START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF], this number is used as an index to switch between imperfect and prefect maintenance actions, however maintenance durations are omitted.

2. Virtual age-based modeling. The second approach links the virtual age of a system to its deterioration level.

When a repair removes a portion of virtual age accumulated since the last repair, it also puts the system back to a deterioration level where it was some time before. By this way, the repair is past-dependent. [START_REF] Ahmadi | A new approach to modeling condition-based maintenance for stochastically deteriorating systems[END_REF][START_REF] Ahmadi | Scheduling preventive maintenance for a nonperiodically inspected deteriorating system[END_REF]; [START_REF] Mercier | On the modelling of imperfect repairs for a continuously monitored gamma wear process through age reduction[END_REF] are a few authors who employ this approach to developed their CBM models. Using the Kijima's type I model [START_REF] Kijima | Some results for repairable systems with general repair[END_REF], [START_REF] Ahmadi | A new approach to modeling condition-based maintenance for stochastically deteriorating systems[END_REF][START_REF] Ahmadi | Scheduling preventive maintenance for a nonperiodically inspected deteriorating system[END_REF] has explained how past-dependent partial repairs affect the system deterioration level and system failure rate. Beyond the impacts on the system deterioration and failure behavior, [START_REF] Mercier | On the modelling of imperfect repairs for a continuously monitored gamma wear process through age reduction[END_REF] have based on the ARA with memory 1 (ARA 1 ) model [START_REF] Doyen | Classes of imperfect repair models based on reduction of failure intensity or virtual age[END_REF] to decide if imperfect repair or perfect replacement is more suitable for a preventive action.

3. Deterioration level-based modeling. Unlike the two above approaches, the third one impacts directly the system deterioration. It assumes that a repair just can return the system to a deterioration level worse than the one existing at the last repair. In this spirit, [START_REF] Ponchet | Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements[END_REF] have mimicked the ARA 1 model to build so-called Arithmetic Reduction of Deterioration with memory 1 (ARD 1 ) model. This model enables a connection to the past in the sense that each repair reduces a part of the deterioration accumulated by the system from the last repair. The ARD 1 model is further implemented by [START_REF] Castro | Performance measures for a deteriorating system subject to imperfect maintenance and delayed repairs[END_REF] to determine whether a repair or a replacement should be carried out at a preventive maintenance time.

For a better choice among the aforementioned approaches, some comparative works have been done. For instance, [START_REF] Mercier | Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system[END_REF] have performed stochastic comparisons between the ARA 1 and ARD 1 models under the assumption of a Gamma deteriorating system. Based on a system subject to the Wiener deterioration process, Kahle

(2019) has recently compared Kijima's type models [START_REF] Kijima | Some results for repairable systems with general repair[END_REF] applied for both the system virtual age and the system deterioration. These models also implies that the higher the repairs number, the more the system is deteriorated.

Consequently, impacts of repairs number are considered indirectly in the second and third approaches. Moreover, it seems that only the latter is able to take into account directly deterioration information revealed by the system monitoring in repair models.

Owing to its advantages, we have applied the deterioration level-based approach to develop our CBM model considering past-dependent partial repairs for a continuously deteriorating system. Compared to previous works using the same approach (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] Castro | Performance measures for a deteriorating system subject to imperfect maintenance and delayed repairs[END_REF][START_REF] Zhao | Optimal inspection and replacement policy based on experimental degradation data with covariates[END_REF]), the developed CBM model has three major differences. Firstly, to express the past dependency of the system deterioration level, we just use a truncated probability distribution. After a repair, the restarting deterioration level of the system is sampled from a probability distribution truncated by the deterioration levels just before a current repair and just after the last repair/replacement.

Unlike arithmetic reduction type and Kijima's type models, this simple model allows breaking the memory assumption:

the system after a repair is put back to an exact deterioration level where it was in the past, which is not easily verified in practice due to the stochastic nature of deterioration process. Secondly, we take into account all the possible the effects of past-dependent partial repairs in an unifying CBM model via restarting deterioration states. There are effects on (i) the selection of a type of maintenance actions (either a partial repair or a perfect replacement), (ii) the maintenance duration, (iii) the deterioration level after a repair, and on (iv) the restarting system deterioration behavior. Finally, we analytically derive the long-run expected maintenance cost rate of the condition-based maintained system (CBMS) using the semi-regenerative theory. Even though this approach has now become rather classical in reliability literature [START_REF] Bérenguer | On the mathematical condition-based maintenance modelling for continuously deteriorating systems[END_REF], its development in the context of past dependency is still meaningful, especially in terms of numerical computation and Monte Carlo simulation.

The remainder of this paper is organized as follows. Section 2 gives a detailed description of the considered CBMS.

Sections 3 and 4 are devoted to the mathematical formulation and validation of the asymptotic behavior and the cost model of the maintained system. The sensitivity studies in Section 5 allow us to assess the effects of past-dependent partial repairs on the economic performances of the maintained system. Finally, some conclusions and perspectives are discussed in Section 6.

Description of the condition-based maintained system

Let consider a single-unit system subject to continuous deterioration such as wear, fatigue, corrosion, crack growth [START_REF] Grall | Continuous-time predictive-maintenance scheduling for a deteriorating system[END_REF]. Such a system may consist of one component or one group of associated components whose deterioration state at time t ≥ 0 can be summarized by a scalar random variable X t , with X 0 = 0. For safety, a high deterioration is unacceptable in engineering practice so that the system is declared as failed whenever its deterioration level exceeds a critical threshold L (i.e., X t ≥ L), even if it is physically running. To prevent or correct the system failure, maintenance actions such as inspection, perfect replacement and partial repair are resorted to. The inspection and replacement are assumed memoryless, while the partial repair is past-dependent via the deterioration level given at the last repair. Since these actions are costly, a deterioration-based maintenance policy with two decision thresholds has been proposed to organize them in a proper manner. A threshold on deterioration levels revealed by inspections, denoted ζ ∈ [0, L), is used to decide wether or not a preventive intervention should be done at a given inspection time. Another threshold on deterioration levels returned by last repairs, denoted η ∈ [0, L], is used to select the type of a preventive action (i.e., partial repair or perfect replacement). Therefore, let E 0 , E 1 , . . . , E j , E j+1 , . . . , be the successive end-of-repair/replacement times, with E 0 = 0, the evolution of the maintained system on the cycle E j , E j+1 , j ∈ N, is as follows.

1. The system is regularly inspected at times

T j,k = E j + k • δ, with k = 1, 2, . . ., until X T j,k ≥ ζ.
Let S j denote the starting time of a maintenance action on E j , E j+1 , j ∈ N, then over E j , S j , X t acts like a homogeneous Gamma process [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF] with shape parameter α 0 + α X E + j and scale parameter β > 0

{X t } E + j ≤t≤S j ∼ HGP α 0 + α X E + j , β . (1)
The first part of the shape parameter α 0 > 0 is a constant characterizing the proper dynamics of the system deterioration. The second part α X E + j , where α (0) = 0, is a continuous increasing non-negative function of X E + j , 0 ≤ X E + j < L, representing the effect of past dependency on the future deterioration dynamics. Consequently, between two times s and t, E j ≤ s < t ≤ S j , the random deterioration increment X t -X s is Gamma distributed with probability density function (pdf)

f α 0 +α X E + j •(t-s),β (x) = β α 0 +α X E + j •(t-s) x α 0 +α X E + j •(t-s)-1 e -βx Γ α 0 + α X E + j • (t -s) • 1 {x≥0} , (2)
and survival function

F α 0 +α X E + j •(t-s),β (x) = Γ α 0 + α X E + j • (t -s) , βx Γ α 0 + α X E + j • (t -s) , (3)
where 1 {•} denotes the indicator function which equals 1 if the argument is true and 0 otherwise,

Γ (α) = ∞ 0 z α-1 e -z dz and Γ (α, x) = ∞ x z α-1 e -z
dz are respectively the complete and upper incomplete gamma functions. The inspection is assumed perfect in the sense that it reveals the exact deterioration level of the system. Moreover, it takes negligible time, has no effect on the system deterioration, and incurs a constant unit cost C m > 0.

At an inspection time

T j,k = E j + k • δ, k = 1, 2, .
. ., a CBM decision rule based on both X T j,k and X E + j is adopted.

(a) If X T j,k ≥ L, a corrective replacement (CR) with constant unit cost C c is carried out immediately on the failed system (i.e., S j = T j,k ). It takes a constant duration λ 0 due to e.g., the material set-up, the system dismantling and reassembly. During the CR, the system is inactivated, so that the system deterioration level keeps unchanged (i.e., X E - j+1 = X S j , with E j+1 = S j + λ 0 ). The system inactivity incurs a constant cost rate C i > 0. Furthermore, before the CR starts, the system has been failed and unavailable until S j . Such a system unavailability incurs a constant cost rate C u > C i because it is unforeseen. After the CR, the system is AGAN

such that X E + j+1 = 0. (b) If ζ ≤ X T j,k < L and X E + j ≥ η, a preventive replacement (PR) with constant unit cost C p < C c
is immediately carried out on the unrepairable badly deteriorated system (i.e., S j = T j,k ). The PR also takes a constant duration λ 0 due to the same reason as above. Over S j , E j+1 , with E j+1 = S j + λ 0 , the system is inactivated at the cost rate C i and it deterioration level is unchanged (i.e., X E - j+1 = X S j ). After the PR, the system is reset to be AGAN (i.e., C p is immediately carried out on the repairable badly deteriorated system (i.e., S j = T j,k ). The PPR requires a duration I j depending on X E + j and X S j such that

X E + j+1 = 0). (c) If ζ ≤ X T j,k < L and X E + j < η, a preventive partial repair (PPR) with constant unit cost C r ∈ C m ,
I j = λ 0 + λ X E + j , X S j , (4)
where λ 0 is a constant duration as for replacements, λ X E + j , X S j is a continuous increasing function of X E + j and X S j , with λ (0, 0) = 0. So, the higher the value of X E + j or X S j , the longer the required duration I j for PPR. As in the case of replacements, the system is inactivated during the PPR at the cost rate C i , and its deterioration level is still unchanged (i.e., X E - j+1 = X S j , with E j+1 = S j + I j ). Just after the PPR, the system restarts with a deterioration level X E + j+1 ∈ X E + j , X S j sampled from a pdf truncated by X S j and

X E + j X E + j+1 ∼ g y | X S j , X E + j . (5) (d) If X T j,k < ζ, no further intervention is needed at T j,k
, and hence nothing is changed. The decision is postponed to the next inspection at T j,k+1 = T j,k + δ.

The next cycle begins at E j+1 with initial deterioration level X E + j+1 . On which, the system deterioration evolves following a homogeneous Gamma process with shape parameter α 0 + α X E + j+1 and scale parameter β > 0. If X E + j+1 ≥ X E + j (it is true when the last maintenance action is a PPR), the system deteriorates with a higher average speed and a higher variance than before. Figure 1 illustrates the deterioration evolution of the CBMS over some first repair/replacement cycles. For the considered system, the failure threshold L, and the maintenance costs C m , C r , C p , C c , C i and C u are input data.

The scalars α 0 , β and λ 0 , and the functions α (•), λ (•) and g (• | •, •) are parameters to be determined from deterioration and maintenance data. Although this problem has not been dealt with in this paper, we still believe that the following twosteps procedure could be applied. For some conjectured parametric forms of α (•), λ (•) and g (• | •, •), classical methods (e.g., maximum likelihood method, method of moment, etc.) are used to estimate the model parameters (including α 0 , β and λ 0 ) from deterioration and maintenance data. Next, we perform goodness-of-fit tests to find the best fit of the data. For this estimation-testing procedure, the deterioration and maintenance data are obviously prerequisite. This is why building such a data-set is recognized as a key perspective of the paper. Finally, the inspection period δ, and the deterioration thresholds ζ and η are decision variables to be jointly optimized. As argued in (Wagner, 1975, chapter 11), the long-run expected cost rate is a suitable objective function

C ∞ (δ, ζ, η) = lim t→∞ C (t) t , (6)
where C (t) denotes the cumulative maintenance cost incurred in the time interval [0, t]. To illustrate the practicalness of the proposed CBMS, let us introduce the gyroscope equipment represented in [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF][START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF]. Gyroscope is a core component in inertial navigation systems. Due to the wear of rotor spin axis and the friction of gimbal bearings, the gyroscopic drift increases over time, and hence degrades the gyroscope performance. Therefore, the drift can be seen as a deterioration index of the gyroscope. In the experiment provided [START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF][START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF] after adjustments and the deterioration speed of the gyroscope, their characteristics is completely covered in our CBMS.

Although the duration of maintenance actions has not been mentioned in this experiment, various existing works (see e.g., [START_REF] Castanier | A sequential condition-based repair/replacement policy with nonperiodic inspections for a system subject to continuous wear[END_REF][START_REF] Liao | Maintenance of continuously monitored degrading systems[END_REF]) confirm that the repair duration is very influenced by partial repairs.

Asymptotic deterioration behavior of the condition-based maintained system

Evaluating the maintenance cost rate in the long term ( 6) requires the study of the asymptotic deterioration behavior of the maintained system. In practice, the system does not wait till infinity to reach its asymptotic behavior, but rather at a finite moment from which the short-run maintenance cost rate converges to the long-run one with an acceptable error. Theoretically, the study of asymptotic behavior can be significantly simplified thanks to the semi-regenerative properties of the deterioration process of the maintained system [START_REF] Bérenguer | On the mathematical condition-based maintenance modelling for continuously deteriorating systems[END_REF]. In this section, after analyzing how these semi-regenerative properties simplifies the evaluation of (6), we derive the stationary law of the maintained system deterioration.

Semi-regenerative properties of the maintained system deterioration

The semi-regenerative properties of the deterioration process {X t } t≥0 of the maintained system allow studying the asymptotic deterioration behavior of the system on a limited horizon instead of an infinite horizon. After the end of a system repair/replacement at E j , j ∈ N, the future deterioration process X t+E j t≥0 of the maintained system depends on its past {X t } 0≤t≤E j only via X E + j . Therefore, apart from its regenerative structure, {X t } t≥0 appears as a semi-regenerative process (Cinlar, 1975, page 343), with repair/replacement end times E + j , j ∈ N, as semi-regenerative (Markov renewal) times (see Figure 1). Embedded in {X t } t≥0 , there exits a Markov chain Y j j∈N , Y j = X E + j , with stationary law π. As shown in [START_REF] Grall | Continuous-time predictive-maintenance scheduling for a deteriorating system[END_REF], the study of asymptotic behavior of {X t } t≥0 can be restricted to a single semi-regenerative cycle (also known as Markov renewal cycle) defined by two successive repair/replacement times, and the long-run maintenance cost rate (6) can be expressed by

C ∞ (δ, ζ, η) = lim t→∞ C (t) t = E π C 0 + , E + 1 E π [∆E 1 ] , (7)
where E π [•] denotes the s-expectation with respect to the stationary π, and ∆E 1 = E 1 denotes the length of the first Markov renewal cycle 0 + , E + 1 . We note that 0 + is not merely the initial working time of the system at which X 0 + = 0, but rather the beginning of a Markov renewal cycle at which X 0 + = x, where 0 ≤ x < L. Some elements of proof of (7) are provided in [START_REF] Mercier | A condition-based imperfect replacement policy for a periodically inspected system with two dependent wear indicators[END_REF]. A short length of ∆E 1 simplifies the analysis of the system deterioration behavior, thence allows an analytical evaluation of C ∞ (δ, ζ, η). However, the price paid to this simplicity is the derivation of the stationary law π of the embedded Markov chain Y j j∈N , which is the main difficulty of this approach.

Stationary law of the embedded Markov chain Y j j∈N

The embedded Markov chain Y j j∈N describes the system deterioration state at the end of a repair/replacement action.

It 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 (see Figure 1). Therefore, there exists a stationary measure π for Y j j∈N on [0, L), which is the solution of the invariance equation [START_REF] Asmussen | Stochastic simulation: algorithms and analysis[END_REF], page 97)

π (dy) = [0,L) P (x, dy) π (dx) , (8)
where P (x, dy) stands for the transition kernel of Y j j∈N from X E + j = x to X E + j+1 = y. In the following, we seek a closed-form expression of P (x, dy), and thence propose a solution for (8).

Transition kernel P (x, dy)

A closed-form expression of P (x, dy) can be obtained by exhaustively analyzing different possible scenarios of the maintained system under the (δ, ζ, η) policy from the beginning E + j to the end E + j+1 of the j-th Markov renewal cycle. As shown in Appendix A, we can express P (x, dy) as follows.

1. when ζ < η,

P (x, dy) = δ 0 (dy) • (ρ 1 (x) + ρ 2 (x)) + (p 1 (y | x) + p 2 (y | x)) • 1 {y 0,x≤y<L} • dy • 1 {0≤x<ζ} + δ 0 (dy) • ρ 1 (x) + p 3 (y | x) • 1 {y 0,x≤y<L} • dy • 1 {ζ≤x<η} + δ 0 (dy) • 1 {η≤x<L} , (9) 2. when η ≤ ζ, P (x, dy) = δ 0 (dy) • (ρ 1 (x) + ρ 2 (x)) + (p 1 (y | x) + p 2 (y | x)) • 1 {y 0,x≤y<L} • dy • 1 {0≤x<η} + (ρ 3 (x) + ρ 4 (x)) • δ 0 (dy) • 1 {η≤x<ζ} + δ 0 (dy) • 1 {ζ≤x<L} , (10)
where, given

X E + j = x,
• ρ 1 (x) denotes the conditional probability of a CR after one inspection period since

E + j ρ 1 (x) = F(α 0 +α(x))δ,β (L -x) , (11)
• ρ 2 (x) denotes the conditional probability of a CR after multiple inspection periods since

E + j ρ 2 (x) = ζ x F(α 0 +α(x))δ,β (L -w) ∞ k=1 f (α 0 +α(x))kδ,β (w -x) dw, ( 12
)
• ρ 3 (x) denotes the conditional probability of a PR after one inspection period since

E + j ρ 3 (x) = F(α 0 +α(x))δ,β (ζ -x) , (13)
• ρ 4 (x) denotes the conditional probability of a PR after multiple inspection periods since

E + j ρ 4 (x) = ζ x F(α 0 +α(x))δ,β (ζ -w) ∞ k=1 f (α 0 +α(x))kδ,β (w -x) dw, (14)
• p 1 (y | x) stands for the conditional pdf of a PPR after one inspection period since E + j , with x ∈ 0, η),

p 1 (y | x) = L ζ g (y | x, r) f (α 0 +α(x))δ,β (r -x) dr, (15)
• p 2 (y | x) stands for the conditional pdf of a PPR after multiple inspection periods since

E + j p 2 (y | x) = ζ x L ζ g (y | x, z) f (α 0 +α(x))δ,β (z -w) dz ∞ k=1 f (α 0 +α(x))kδ,β (w -x) dw, ( 16
)
• p 3 (y | x) stands for the conditional pdf of a PPR after one inspection period since E + j , with x ∈ η, ζ),

p 3 (y | x) = L x g (y | x, r) f (α 0 +α(x))δ,β (r -x) dr, ( 17
) in which f (α 0 +α(x))•(•),β (•) is given from (2) and F(α 0 +α(x))δ,β (•) is derived from (3).
We note that the expression of P (x, dy) consists of both the dirac part and continuous part. The magnitude of the Dirac measure located at 0 represents the probability of a replacement (either corrective or preventive) done during a Markov renewal cycle. Besides, P (x, dy) does not impacted by the maintenance duration, in which the system deterioration keeps unchanged.

Solution for the stationary law π (dy)

As the expression of P (x, dy), the stationary law π (dy) is also a convex combination of Dirac mass function and a continuous pdf. Appendix B gives the mathematical expression of π (dy) as follows.

1. When ζ < η,

π (dy) = a • δ 0 (dy) + (1 -a) • b 1 (y) • 1 {0<y<ζ} dy + (1 -a) • b 2 (y) • 1 {ζ≤y<η} dy + (1 -a) • b 3 (y) • 1 {η≤y<L} dy, ( 18
)
where

a = 1 1 + ζ 0 B 1 (y) dy + η ζ B 2 (y) dy + L η B 3 (y) dy , (19)
and

b 1 (y) = a 1 -a • B 1 (y) , b 2 (y) = a 1 -a • B 2 (y) , and b 3 (y) = a 1 -a • B 3 (y) . ( 20
)
B 1 (y), B 2 (y) and B 3 (y) are computed by

• when 0 < y < ζ, B 1 (y) = p 1 (y | 0) + p 2 (y | 0) + y 0 B 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx, ( 21
) • when ζ < y < η, B 2 (y) = p 1 (y | 0) + p 2 (y | 0) + ζ 0 B 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx + y ζ B 2 (x) • p 3 (y | x) dx, (22)
• when η < y < L,

B 3 (y) = p 1 (y | 0) + p 2 (y | 0) + ζ 0 B 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx + η ζ B 2 (x) • p 3 (y | x) dx, ( 23
)
where p 1 (y | x), p 2 (y | x) and p 3 (y | x) are given from ( 15), ( 16) and (17).

When

η ≤ ζ, π (dy) = c • δ 0 (dy) + (1 -c) • d 1 (y) • 1 {0<y<η} dy + (1 -c) • d 2 (y) • 1 {η≤y<ζ} dy + (1 -c) • d 3 (y) • 1 {ζ≤y<L} dy, ( 24
)
where

c = 1 1 + η 0 D 1 (y) dy + ζ η D 3 (y) dy + L ζ D 3 (y) dy , (25)
and

d 1 (y) = c 1 -c • D 1 (y) , d 2 (y) = c 1 -c • D 2 (y) , and d 3 (y) = c 1 -c • D 3 (y) . ( 26
)
D 1 (y), D 2 (y) and D 3 (y) are computed by

• when 0 < y < η D 1 (y) = p 1 (y | 0) + p 2 (y | 0) + y 0 D 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx, ( 27
) • when η < y < ζ D 2 (y) = p 1 (y | 0) + p 2 (y | 0) + η 0 D 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx, ( 28
) • when ζ < y < L D 3 (y) = p 1 (y | 0) + p 2 (y | 0) + η 0 D 1 (x) • (p 1 (y | x) + p 2 (y | x)) dx, ( 29
)
where p 1 (y | x) and p 2 (y | x) are given from ( 15) and ( 16).

Solving the non-homogeneous linear Volterra integral equations of the second kind ( 21), ( 22) and ( 27) allows to fully derive π (dy). However, their analytical solutions are not easy to handle. To overcome this obstacle, the Heun' s numerical method (Kharab and Guenther, 2011, pages 334-335) is used to approximate the solution of ( 21), ( 22) and ( 27). Appendix B.2 gives the detail of this numerical method.

Cost-based optimization of the condition-based maintained system

This section aims at optimizing the considered CBMS using the long-run maintenance cost rate (6). To this end, we formulate a closed-form expression of C ∞ (δ, ζ, η) following ( 7). Next, we apply derivative free algorithms for blackbox optimization (e.g., generalized pattern search [START_REF] Audet | Derivative-Free and Blackbox Optimization[END_REF]) to C ∞ (δ, ζ, η) to search the optimal decision parameters δ opt , ζ opt , η opt .

Maintenance cost rate evaluation

From the decision structure implemented in the considered CBMS, we can express the cumulative cost incurred in the time interval [0, t] as

C (t) = C r • N r (t) + C p • N p (t) + C c • N c (t) + C m • N m (t) + C u • U (t) + C i • I (t) , (30)
where N r (t), N p (t), N c (t) and N m (t) denote respectively the number of PPR, of PR, of CR, and of inspections in [0, t], U (t) and I (t) stand for total duration of system unavailability and system inactivity in [0, t]. Using (7), we obtain the 31) can be rewritten as

expression of C ∞ (δ, ζ, η) as C ∞ (δ, ζ, η) = E π C 0 + , E + 1 E π [∆E 1 ] = C r • E π N r 0 + , E + 1 E π [∆E 1 ] + C p • E π N p 0 + , E + 1 E π [∆E 1 ] + C c • E π N c 0 + , E + 1 E π [∆E 1 ] + C m • E π N m 0 + , E + 1 E π [∆E 1 ] + C u • E π U 0 + , E + 1 E π [∆E 1 ] + C i • E π I 0 + , E + 1 E π [∆E 1 ] . (31) Since 0 + , E + 1 = 0 + , S 1 ∪ S 1 , E + 1 , where S 1 = δ • N m 0 + , E + 1 , (
C ∞ (δ, ζ, η) = 1 δ • E π N m 0 + , E + 1 + E π I 0 + , E + 1 • C r • E π N r 0 + , E + 1 + C p • E π N p 0 + , E + 1 +C c • E π N c 0 + , E + 1 + C m • E π N m 0 + , E + 1 + C u • E π U 0 + , E + 1 + C i • E π I 0 + , E + 1 . (32)
Hereinafter, we analyze all the possible maintenance scenarios on the first Markov renewal cycle 0 + , E + 1 , and thence we derive mathematical expressions of

E π N r 0 + , E + 1 , E π N p 0 + , E + 1 , E π N c 0 + , E + 1 , E π N m 0 + , E + 1 , E π U 0 + , E + 1 and E π I 0 + , E + 1 .
The exactness of the formulation is also justified by numerical experiments.

Possible maintenance scenarios on the first Markov renewal cycle

Let consider the Markov renewal cycle 0 + , E + 1 with X 0 + = y, the decision structure of (δ, ζ, η) policy leads to following possible maintenance scenarios.

1. When ζ < η, then (a) the system maintenance starts after one inspection period δ since 0 + (i.e., S 1 = δ) by

• scenario 1: a PPR with duration λ 0 + λ (X 0 + , X δ ) if {0 ≤ X 0 + < ζ ≤ X δ < L} or {ζ ≤ X 0 + < η, X δ < L},
• scenario 2: a PR with duration λ 0 if {η ≤ X 0 + < X δ < L},

• scenario 3: a CR with duration λ 0 if {0 ≤ X 0 + < L ≤ X δ }, (b) the system maintenance starts after a multiple of inspection period (k + 1) δ, k = 1, 2, . . ., since 0 + (i.e.,

S 1 = (k + 1) δ) by • scenario 4: a PPR with duration λ 0 + λ X 0 + , X (k+1)δ if 0 ≤ X 0 + ≤ X kδ < ζ ≤ X (k+1)δ < L , • scenario 5: a CR with duration λ 0 if 0 ≤ X 0 + ≤ X kδ < ζ < L ≤ X (k+1)δ .
2. When η ≤ ζ, then (a) the system maintenance starts after one inspection period δ since 0 + (i.e., S 1 = δ) by

• scenario 6: a PPR with duration λ 0 + λ (X 0 + , X δ ) if {0 ≤ X 0 + < η ≤ ζ ≤ X δ < L}, • scenario 7: a PR with duration λ 0 if {η ≤ X 0 + < ζ ≤ X δ < L} or {ζ ≤ X 0 + < X δ < L}, • scenario 8: a CR with duration λ 0 if {0 ≤ X 0 + < L ≤ X δ },
(b) the system maintenance starts after a multiple of inspection period (k + 1) δ, k = 1, 2, . . ., since 0 + (i.e.,

S 1 = (k + 1) δ) by • scenario 9: a PPR with duration λ 0 + λ X 0 + , X (k+1)δ if 0 ≤ X 0 + ≤ X k•δ < ζ ≤ X (k+1)δ < L • scenario 10: a PR with duration λ 0 if η ≤ X 0 + < X kδ < ζ ≤ X (k+1)δ < L , • scenario 11: a CR with duration λ 0 if 0 ≤ X 0 + < η, X kδ < ζ < L ≤ X (k+1)δ or η ≤ X 0 + < X kδ < ζ < L ≤ X (k+1)δ .
The above scenarios are the basis to compute the required expectations.

Expected number of preventive partial repairs over the first Markov renewal cycle

As shown in Appendix C, we can express the expected value of N r 0 + , E + 1 with respect to the stationary law π as 1. when ζ < η,

E π N r 0 + , E + 1 = a •         Fα 0 δ,β (ζ) -Fα 0 δ,β (L) + ζ 0 Fα 0 δ,β (ζ -w) -Fα 0 δ,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -a) • ζ 0 F(α 0 +α(y))δ,β (ζ -y) -F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (ζ -w) -F(α 0 +α(y))δ,β (L -w) × ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         b 1 (y) dy + (1 -a) • η ζ F (α 0 +α(y))δ,β (L -y) b 2 (y) dy, (33)
where a, b 1 (y) and b 2 (y) are given from ( 19) and ( 20

); 2. when η ≤ ζ, E π N r 0 + , E + 1 = c •         Fα 0 δ,β (ζ) -Fα 0 δ,β (L) + ζ 0 Fα 0 δ,β (ζ -w) -Fα 0 δ,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -c) • η 0 F(α 0 +α(y))δ,β (ζ -y) -F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (ζ -w) -F(α 0 +α(y))δ,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 1 (y) dy, (34)
where c and d 1 (y) are given from ( 25) and (26).

Expected number of preventive replacements over the first Markov renewal cycle

As shown in Appendix D, the expected value of N p 0 + , E + 1 with respect to the stationary law π is given by 1. when ζ < η,

E π N p 0 + , E + 1 = (1 -a) • L η F (α 0 +α(y))δ,β (L -y) b 3 (y) dy, ( 35
)
where a and b 3 (y) are given from ( 19) and (20);

2. when η ≤ ζ,

E π N p 0 + , E + 1 = (1 -c) • ζ η F(α 0 +α(y))δ,β (ζ -y) -F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (ζ -w) -F(α 0 +α(y))δ,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 2 (y) dy + L ζ F (α 0 +α(y))δ,β (L -y) d 3 (y) dy , ( 36
)
where c, d 2 (y) and d 3 (y) are given from ( 25) and ( 26).

Expected number of corrective replacements over the first Markov renewal cycle

Appendix E gives the expected value of N c 0 + , E + 1 with respect to the stationary law π as 1. when ζ < η,

E π N c 0 + , E + 1 = a •         Fα 0 δ,β (L) + ζ 0 Fα 0 δ,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -a) ×         ζ 0         F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         b 1 (y) dy
E π N c 0 + , E + 1 = c •         Fα 0 δ,β (L) + ζ 0 Fα 0 δ,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -c) ×         η 0         F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 1 (y) dy + ζ η         F(α 0 +α(y))δ,β (L -y) + ζ y F(α 0 +α(y))δ,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 2 (y) dy + L ζ F(α 0 +α(y))δ,β (L -y) d 3 (y) dy , (38)
where a, d 1 (y), d 2 (y) and d 3 (y) are given from ( 25) and (26).

Expected number of inspections over the first Markov renewal cycle

Based on the expectations of the scenarios derived in Appendix C, Appendix D and Appendix E, the expected value of N m 0 + , E + 1 with respect to the stationary law π is obtained by

1. when ζ < η, E π N m 0 + , E + 1 = a • Fα 0 δ,β (ζ) + (1 -a) • ζ 0 F(α 0 +α(y))δ,β (ζ -y) b 1 (y) dy + η ζ b 2 (y) dy + L η b 3 (y) dy + a • ζ 0 Fα 0 δ,β (ζ -w)         ∞ k=1 (k + 1) f α 0 kδ,β (w)         dw + (1 -a) × ζ 0         ζ y F(α 0 +α(y))δ,β (ζ -w)         ∞ k=1 (k + 1) f (α 0 +α(y))kδ,β (w -y)         dw         b 1 (y) dy, (39)
where a, b 1 (y), b 2 (y) and b 3 (y) are given from ( 19) and ( 20

); 2. when η ≤ ζ, E π N m 0 + , E + 1 = c • Fα 0 δ,β (ζ) + (1 -c) • η 0 F(α 0 +α(y))δ,β (ζ -y) d 1 (y) dy+ ζ η F(α 0 +α(y))δ,β (ζ -y) d 2 (y) dy + L ζ d 3 (y) dy + c • ζ 0 Fα 0 δ,β (ζ -w)         ∞ k=1 (k + 1) f α 0 kδ,β (w)         dw + (1 -c) • η 0         ζ y F(α 0 +α(y))δ,β (ζ -w)         ∞ k=1 (k + 1) f (α 0 +α(y))kδ,β (w -y)         dw         d 1 (y) dy + (1 -c) • ζ η         ζ y F(α 0 +α(y))δ,β (ζ -w)         ∞ k=1 (k + 1) f (α 0 +α(y))kδ,β (w -y)         dw         d 2 (y) dy, (40)
where a, d 1 (y), d 2 (y) and d 3 (y) are given from ( 25) and (26).

Expected duration of the system unavailability over the first Markov renewal cycle

Appendix F shows the expected value of U 0 + , E + 1 with respect to the stationary law π as 1. when ζ < η,

E π U 0 + , E + 1 = δ 0         a •         Fα 0 t,β (L) + ζ 0 Fα 0 t,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -a) •         ζ 0         F(α 0 +α(y))t,β (L -y) + ζ y F(α 0 +α(y))t,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         b 1 (y) dy + η ζ F(α 0 +α(y))t,β (L -y) b 2 (y) dy + L η F(α 0 +α(y))t,β (L -y) b 3 (y) dy dt, (41)
where a, b 1 (y), b 2 (y) and b 3 (y) are given from ( 19) and ( 20

); 2. when η ≤ ζ, E π U 0 + , E + 1 = δ 0         c •         Fα 0 t,β (L) + ζ 0 Fα 0 t,β (L -w) ∞ k=1 f α 0 kδ,β (w) dw         + (1 -c) •         η 0         F(α 0 +α(y))t,β (L -y) + ζ y F(α 0 +α(y))t,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 1 (y) dy + ζ η         F(α 0 +α(y))t,β (L -y) + ζ y F(α 0 +α(y))t,β (L -w) ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         d 2 (y) dy + L ζ F(α 0 +α(y))t,β (L -y) d 3 (y) dy dt, (42)
where a, d 1 (y), d 2 (y) and d 3 (y) are given from ( 25) and (26).

Expected duration of the system inactivity over the first Markov renewal cycle

As proved in Appendix G, the expected duration of the system inactivity E π I 0 + , E + 1 can be computed by

E π I 0 + , E + 1 = λ 0 + E π I 1 0 + , E + 1 , (43)
in which 1. when ζ < η,

E π I 1 0 + , E + 1 = a •         L ζ λ (0, w) f α 0 δ,β (w) dw + ζ 0 L ζ λ (0, v) f α 0 δ,β (v -w) dv ∞ k=1 f α 0 kδ,β (w) dw         + (1 -a) × ζ 0         L ζ λ (y, w) f (α 0 +α(y))δ,β (w -y) dw + ζ y L ζ λ (y, v) f (α 0 +α(y))δ,β (v -w) dv ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         × b 1 (y) dy + (1 -a) • η ζ L y λ (y, w) f (α 0 +α(y))δ,β (w -y) dw b 2 (y) dy, ( 44
)
where a, b 1 (y) and b 2 (y) are given from ( 19) and (20);

2. when η ≤ ζ,

E π I 1 0 + , E + 1 = c •         L ζ λ (0, w) f α 0 δ,β (w) dw + ζ 0 L ζ λ (0, v) f α 0 δ,β (v -w) dv ∞ k=1 f α 0 kδ,β (w) dw         + (1 -c) × η 0         L ζ λ (y, w) f (α 0 +α(y))δ,β (w -y) dw + ζ y L ζ λ (y, v) f (α 0 +α(y))δ,β (v -w) dv ∞ k=1 f (α 0 +α(y))kδ,β (w -y) dw         × d 1 (y) dy, ( 45
)
where c and d 1 (y) are given from ( 25) and (26).

Maintenance cost model validation

To validate the above mathematical formulation, we effectuate numerical comparisons between the results given by the numerical computation and the Monte Carlo simulation of δ, ζ, η). For the numerical computation, we use the well-known trapezoidal rule to approximate integrals in the considered expectations. We also propose a simple way to derive the simulated results by Monte Carlo approach in Appendix H. In the following, an illustration is given on the basis of the maintained system defined by the set of parameters L = 15, 1. The almost identical results given by both the approaches justify the correctness of the developed mathematical cost model.

E π N r 0 + , E + 1 , E π N p 0 + , E + 1 , E π N c 0 + , E + 1 , E π N m 0 + , E + 1 , E π U 0 + , E + 1 , E π I 0 + , E + 1 , as well as of C ∞ (
α 0 = 1, α X E + j = 0.1 • X E + j , β = 1, λ 0 = 1, λ X E + j , X S j = 0.1 • X E + j + 0.2 • X S j ,

Optimum existence and searching

Optimizing the (δ, ζ, η) policy is to seek the triplet of decision parameters δ opt , ζ opt , η opt that minimizes C ∞ (δ, ζ, η) cannot cover all possible configurations, it is still an acceptable solution when analytical approach is impossible. A general conclusion drawn from these observations is that each of decision parameters δ, ζ and η has its own effect on the maintenance cost rate, and they have to be jointly optimized to achieve the best performance for the (δ, ζ, η) policy. As an illustration, we sketch in 4)). The generalized pattern search algorithm allows to find quickly the optimal configuration of the (δ, ζ, η) policy.

C ∞ δ opt , ζ opt , η opt = min (δ,ζ,η) {C ∞ (δ, ζ, η) , δ > 0, 0 ≤ ζ ≤ L, 0 ≤ η ≤ L} . (46) Config. Approach E π [N r ] E π N p E π [N c ] E π [N m ] E π [U] E π [I] C ∞ (δ, ζ,

Numerical assessment of the condition-based maintained system

This section aims at using numerical experiments to confirm the effectiveness of the proposed CBMS, and to understand more deeply the impacts of the past-dependent PPR on the economic performance of the maintained system.

To this end, we perform comparative studies of optimal long-run cost rate between the considered system (δ, ζ, η) and its two extreme cases (i.e., system with pure PR (δ, ζ, η = 0) (see e.g., [START_REF] Huynh | A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events[END_REF])), and system with pure PPR (δ, ζ, η = L) (see e.g., [START_REF] Meier-Hirmer | Maintenance optimization for a system with a gamma deterioration process and intervention delay: application to track maintenance[END_REF])). Numerous experiments have been done for divers configurations of system characteristics and maintenance costs. However, illustrations shown in this section are just given from the maintained system characterized by

• a Gamma deterioration process with linear shape parameter HGP α

0 + α X E + j , β = HGP α 0 + α 1 • X E + j , β , • a linear duration for PPR λ 0 + λ X E + j , X S j = λ 0 + λ 1 • X E + j + λ 2 • X S j ,
• a continuous uniform pdf for g (y | x, r), and maintenance cost (i.e., C p ) will be stated latter depending on specific sensitivity studies.

where α 0 = 1, β = 1, λ 0 = 1, λ 2 = 0.

Economic performance of the condition-based maintained system

The distinction in economic performance of the three considered maintained systems comes from the difference between the PR cost C p and the PPR cost C r . Therefore, to see how good the system (δ, ζ, η) is, we vary the cost ratio C p C r in a wide range (C r has been already fixed), and compare its optimal long-run cost rates with the other systems. The result shown in Figure 5 is obtained when fixing α 1 = 0.1, λ 1 = 0.25 and varying C p C r from 3 to 15 with step 0.5. The figure shows clearly that the system (δ, ζ, η) always saves more maintenance cost and returns to either the system (δ, ζ, η = 0) or the system (δ, ζ, η = L) in worse cases. Indeed, it is equivalent to the former when C p is relatively small, and to the latter when C p is very high. The system (δ, ζ, η) reaches its best profit at a medium value of the ratio C p C r . From the economic aspect, this result confirms that there is no risk in using the proposed CBMS compared to more classical ones.

Impacts of the past-dependent preventive partial repairs

To better understand the impacts of past-dependent PPR on the economic performance of the three maintained systems, we study separately how α 1 and λ 1 affect the evolution their optimal long-run maintenance cost rate. Consequently, two following configurations have been considered 1. α 1 is varied from 0 to 0.2 with step 0.025, and λ 1 is fixed at 0, 2. α 1 is fixed at 0, and λ 1 is varied from 0 to 0.9 with step 0.1.

We note that only one impact of the past-dependent PPR (either on the system deterioration dynamics via α 1 or on the repair duration via λ 1 ) is taken into account in each above configuration. The increasing α 1 and λ 1 imply the more and more important impacts of the past-dependent PPR. The PR cost is assumed fixed at C p = 100. The results for the two above configurations are reported in Figures 6a and6b Obviously, the optimal long-run cost rate of the system (δ, ζ, η = 0) is constant for both the considered configurations, because its maintenance decision structure is independent of PPR. Whereas, using the past-dependent partial repairs as a preventive action, the system (δ, ζ, η) and the system (δ, ζ, η = L) incur higher maintenance cost due to the increasing of α 1 and λ 1 . However, looking at the growth of their optimal long-run cost rate, the former resists the negative effects of the PPR much better than the latter. In other words, these negative effects can be significantly reduced if we combine properly the PPR and the PR into a CBM decision rule.

Conclusions and perspectives

The main focus of this paper is to model and evaluate the impacts of the past-dependent PPR on the economic performance of a condition-based maintained deteriorating system. A complete procedure, including the system deterioration modeling, the maintenance effects modeling, the elaboration of CBM decision rule, the formulation and optimization of mathematical cost model, and the maintenance model assessment, has been performed. Numerous numerical experiments confirm that the negative effects of the past-dependent PPR on the economic performance of the maintained system are unavoidable, but can be significantly reduced by coordinating the PPR and the PR into a CBM decision rule. The deterioration-based maintenance policy developed in this paper could be a good candidate. In fact, there is no risk when using the proposed CBMS, because it achieves at least the same cost-savings as the systems with pure PPR or with pure PR.

Given encouraging theoretical results, our next work is to valid the proposed CBM model with real-world data.

A phase of data analysis, parameters estimation, and of models selection for system deterioration process and pastdependent repairs will be implemented before going further with the CBM decision rule. Currently, only the actual system deterioration state is used to make a maintenance decision. Meanwhile, the growing development of prognostics and health management techniques allows us to further access the information about the future system deterioration state [START_REF] Lee | Prognostics and health management design for rotary machinery systems-reviews, methodology and applications[END_REF]. So, one of our perspectives is to study how this kind of information can be integrated in the CBM model to enable maintenance cost reduction. Using a prognostic-based inspection scheme instead of a periodic one could be an improved idea for the considered CBMS. Another perspective is to develop joint models of CBM and spare parts ordering for the considered system. This will remedy a strong assumption in the present CBMS that spare parts are always available for replacement actions.

Figure 1 :

 1 Figure 1: Schematic evolution of the maintained system state

  Figure2: Drift data of two maintained gyroscopes adapted from[START_REF] Hu | A new remaining useful life estimation method for equipment subjected to intervention of imperfect maintenance activities[END_REF][START_REF] Pei | A multi-stage wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities[END_REF]

  +α(y))δ,β (Ly) b 2 (y) dy + L η F(α 0 +α(y))δ,β (Ly) b 3 (y) dy , (37) where a, b 1 (y), b 2 (y) and b 3 (y) are given from (19) and (20); 2. when η ≤ ζ,

  and a continuous uniform pdf for g (y | x, r), under two configurations of the (δ, ζ, η) policy• configuration 1 (ζ < η): δ = 4, ζ = 7, η = 11, • configuration 2 (η ≤ ζ): δ = 4, η = 7, ζ = 11.The set of maintenance costs is chosen as C m = 5, C r = 10, C p = 100, C c = 150, C i = 5 and C u = 25. The duration T for Monte Carlo simulation (see Appendix H) is 10 8 time units. The results are shown as in Table

Figure 3 :

 3 Figure 3: Shapes of C ∞ (δ, ζ, η)

  25, and L = 15. The applied maintenance costs are C m = 5, C r = 10, C c = 150, C i = 10 and C u = 25. The choice of these values are completely arbitrary. The values of other parameters (i.e., α 1 , λ 1 )

Figure

  Figure 5: Sensitivity to Cp Cr

  .

Figure 6 :

 6 Figure 6: Impacts of the past-dependent preventive partial repairs

  pdf of the system deterioration level after a past-dependent PPRC m , C r , C p , C c , C i , C u inspection cost,PPR cost, PR cost, CR cost, inactivity cost rate, unavailability cost rate C (t), C ∞ cumulative maintenance cost up to time t, long-run maintenance cost rate N m (t), N m 0 + , E +

	ζ		threshold for triggering a PR or a PPR
	η		threshold for choosing between a PR and a PPR
	δ		inspection period
	λ 0 , λ (•, •)		constant duration for a replacement, additional deterioration-dependent duration for PPR
	g (• | •, •)
	ARA 1 arithmetic reduction of age with memory 1
	ARD 1 arithmetic reduction of deterioration with memory 1
	AGAN as-good-as-new
	CBM	condition-based maintenance
	CBMS condition-based maintained system(s)
	CR	corrective replacement(s)
	PR	preventive replacement(s)
	PPR	preventive partial repair(s)
	pdf	probability density function(s)
	TBM	time-based maintenance
	Notations
	X t		system deterioration level at time t
	E j , S j		end time of the j-th repair/replacement, starting time of the ( j + 1)-st repair/replacement
	T j,k		k-th inspection time over the j-th repair/replacement cycle E j , E j+1
	α 0 , α X E + j	constant part and X E + j -dependent part of shape parameter of the deterioration process {X t } E + j ≤t≤S j
	β		constant scale parameter of {X t } t≥0
	Γ (•), Γ (•, •) complete Gamma function, upper incomplete Gamma function
	f •,• , F•,•		probability density function, survival function of X t
	L		system failure threshold

Table 1 :

 1 Results for validation the maintenance cost model Analytical proof of optimum existence for the (δ, ζ, η) policy is unfeasible due to the complexity of the mathematical expression of C ∞ (δ, ζ, η). To remedy this obstacle, we propose observing the shapes of C ∞ (δ, ζ, η) when δ, ζ and η vary in a wide rank. Repeating observations for various configurations of system characteristics (i.e., for different α 0 , α X E + j , β, L, g X R + j , λ 0 and λ X E + j , S j ) and maintenance costs (i.e., C m , C r , C p , C c , C i and C u ) allows us to confirm the existence of δ opt , ζ opt , η opt . Even if this approach

	η)

  The triplet δ opt , ζ opt , η opt and the associated cost rate C ∞ δ opt , ζ opt , η opt can be found by the generalized pattern search algorithm[START_REF] Audet | Derivative-Free and Blackbox Optimization[END_REF]. Indeed, continue with the above example, we obtain δ opt = 2.75, ζ opt = 8.75, η opt = 6 and C ∞ δ opt , ζ opt , η opt = 8.2955 when applying the patternsearch solver of Matlab's Global Optimization Toolbox to (32) (see Figure(

			δ opt = 2.75, ζ opt = 8.75, η opt = 6
		15
	δ, ζ, η	5 10	6.00 initial optimal	2.75	11.00	8.75	9. 00 9.00	6.00
		0
			δ		ζ		η
				C ∞ (δ opt , ζ opt , η opt ) = 8.2955
		11
		10
	C ∞
		9
		0 8		8			16	24
					Iteration number
	Figure 4: Optimization with Matlab's patternsearch solver

Acknowledgment

The author would like to thank the three anonymous reviewers for their constructive and valuable comments which have greatly helped in the revision of an earlier version of the article.


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1 Title: POFUT1 and PLAGL2 gene pair linked by a bidirectional promoter: the two in one of tumour progression in colorectal cancer?

Agnès GERMOT, Abderrahman MAFTAH * Limoges University, PEIRENE, EA 7500, Glycosylation and cell differentiation, F-87000 Limoges, France * Corresponding author (abderrahman.maftah@unilim.fr) Colorectal cancer (CRC) is a major health burden that can be well managed if the tumour is detected at an early stage. To identify novel biomarkers for early diagnosis, an increasing number of studies have focused on gene expression dysregulations in the colon and rectal cancers, as in CRC cell lines. Several studies revealed that the 20q11.21 chromosomal region is amplified in at least 65% of CRC cases [START_REF] Carvalho | Multiple putative oncogenes at the chromosome 20q amplicon contribute to colorectal adenoma to carcinoma progression[END_REF]. Such a genomic instability leading to amplification is also found in numerous other cancer types, including breast cancer, lung cancer, melanoma, and numerous others [START_REF] Lefort | Human embryonic stem cells reveal recurrent genomic instability at 20q11.21[END_REF]. The shortest amplified portion associated with acute myeloid leukemia only contains four genes including pleomorphic adenoma gene-like 2 (PLAGL2) and protein O-fucosyltransferase 1 (POFUT1) [START_REF] Mackinnon | The paradox of 20q11.21 amplification in a subset of cases of myeloid malignancy with chromosome 20 deletion[END_REF]. Their expressions have been particularly investigated in the context of CRC by the team of Xiaorong Li, published in the July issue of EBioMedicine [START_REF] Li | et PLAGL2 and POFUT1 are regulated by an evolutionarily conserved bidirectional promoter and are collaboratively involved in colorectal cancer by maintaining stemness[END_REF].

PLAGL2 is a zinc-finger transcription factor located in the nucleus, whose expression shown to be significantly higher in CRC tissues than in adjacent non-tumour tissues [START_REF] Liu | The role of pleomorphic adenoma gene-like 2 in gastrointestinal cancer development, progression, and prognosis[END_REF]. In intestinal organoid cultures, PLAGL2 has shown to drive intestinal epithelial stem cell expression signature, which then activates WNT gene expression and enhances TCF/LEF reporter. Overall, PLAGL2 overexpression in CRC promotes in vitro and in vivo cell proliferation by the Wnt/β-catenin pathway [START_REF] Li | Overexpressed PLAGL2 transcriptionally activates Wnt6 and promotes cancer development in colorectal cancer[END_REF].

POFUT1 is an ER-resident glycosyltransferase that allows fucose addition on S or T included in the C2X4(S/T)C3 consensus motif, where C2 and C3 are the second and third cysteines out of six conserved in EGF-like domain. The major known target of POFUT1 is NOTCH receptor, which contains 14 to 20 O-fucosylation consensus sites according to the four human paralogs; most of the 20 sites on NOTCH1 are O-fucosylated [START_REF] Kakuda | Deciphering the Fringe-mediated Notch code: identification of activating and inhibiting sites allowing discrimination between ligands[END_REF]. POFUT1 is overexpressed in colorectal tumours compared to non-tumour adjacent tissues and positively regulates colorectal tumour progression through activation of Notch signaling pathway [START_REF] Du | POFUT1 promotes colorectal cancer development through the activation of Notch1 signaling[END_REF][START_REF] Chabanais | POFUT1 as a promising novel biomarker of colorectal cancer[END_REF].

As in both PLAGL2 and POFUT1, their overexpression in CRC can be associated to copy number amplification. In the article by Li and colleagues [START_REF] Li | et PLAGL2 and POFUT1 are regulated by an evolutionarily conserved bidirectional promoter and are collaboratively involved in colorectal cancer by maintaining stemness[END_REF], several experimental data are given to support the hypothesis that their correlated expression is driven by the activity of a bidirectional promoter. First, the authors showed that the adjacent head-to-head genes are separated by a short promoter sequence that is evolutionarily conserved in mammals. In 2015, the downregulation of Pofut1 in Plagl2-/-mice had already led Chen and colleagues [START_REF] Chen | Noncanonical role of transferrin receptor 1 is essential for intestinal homeostasis[END_REF] to consider that the two genes likely share a common promoter region. Interestingly, the conservation of the bidirectional promoter would mean that under physiological conditions, 2 coordinated regulation of both genes' expression is essential, at least in mammals. This opens the possibility to reproduce the amplification of the two genes in CRC animal models i.e. mouse and to test the efficiency of modification of the transcription factor binding sites to attenuate the overexpression of the two genes and thus carcinogenicity, without altering their essential physiological functions [START_REF] Li | et PLAGL2 and POFUT1 are regulated by an evolutionarily conserved bidirectional promoter and are collaboratively involved in colorectal cancer by maintaining stemness[END_REF]. Next, the demonstration of the bidirectional activity of the promoter was done in vitro using reporter gene constructs and CRC cell lines. Upon silencing PLAGL2, POFUT1 or both in HCT116 and SW480 cells the authors were able to show that downregulation of the gene pair synergistically reduced colorectal tumourigenesis. Further studies will be required to understand the mechanism leading to this co-expression and biological implications within the tumour environment. Indeed, the tumour tissue is complex because of its cellular heterogeneity compared to a cell line. Tumour is certainly not subjected to the same potential epigenetic regulations, nor to the expression of the same transcription factors. Another interesting finding is that the correlated overexpression of PLAGL2 and POFUT1 initiates deregulation of the two signaling pathways; Wnt/β-catenin and Notch, and subsequently a deregulation of the cell cycle and tumour progression [START_REF] Li | et PLAGL2 and POFUT1 are regulated by an evolutionarily conserved bidirectional promoter and are collaboratively involved in colorectal cancer by maintaining stemness[END_REF]. Further studies are needed to specify the respective contributions of Wnt/β-catenin and Notch signaling in promoting CRC and in maintaining the stemness of CRC stem cells. This study also raises the question of whether, for other cancers, the two genes cooperate in the same way as in the CRC. While it is fairly easy to imagine that overexpression of a transcription factor, like PLAGL2, may modulate several gene expressions, it is a bit more complicated to figure out how overexpression of POFUT1 encoding an O-fucosyltransferase, modifies the expression of genes that affect the cell cycle and the cellular proliferation and invasion. It is widely accepted that O-fucosylation of Notch, by POFUT1, affects Notch signaling. However, it remains to be determined what is the O-fucosylation state of NOTCH receptors in the colorectal tumour. Moreover, NOTCH receptors are not the only targets of POFUT1 and it is not excluded that other O-fucosylated actors participate in the consequences of POFUT1overexpression in CRC.

Finally, this work contributes to the understanding of the mechanisms involved in tumour progression and to the identification of potential biomarkers. In connection with human health, it is rather conceivable to use this type of data for the purpose of improving diagnostics, but it seems more premature to project on a therapeutic approach focusing on the bidirectional promoter region.

Declaration of Competing Interest

The authors declare no conflict of interest.


Michel Simonneau
email: michel.simonneau@ens-paris-saclay.fr

Translational research identifies a metabolism pathway involved in first-episode of schizophrenia: Towards precision medicine

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

background that impacts early brain development and is expressed as a combination of psychotic symptoms -such as hallucinations, delusions and disorganization -and motivational and cognitive dysfunctions (1). The mean lifetime prevalence of the disorder is just below 1%. The first episode of schizophrenia (FESZ) typically occurs in the late teenage years or the early 20s and early diagnosis and treatment may prevent social disability later. At the time of these first psychotic symptoms, neurobiological processes underlying schizophrenia have already been ongoing for many years. The identification of key pathophysiological pathways to the clinical picture of FESZ is needed if we aim to intervene before the window of opportunity is closed and deviations in the brain have become hard-wired [START_REF] Kahn | The neurobiology and treatment of first-episode schizophrenia[END_REF].

In an article in EBioMedicine, Ohnishi and colleagues demonstrate for the first time that betaine metabolism pathway is involved in psychiatric pathophysiology and the betaine potential as a novel psychotherapeutic (3). This article is a translational approach based on exciting results obtained by Koike et al. (4). They used capillary electrophoresis time-of-flight mass spectrometry (CE-TOFMS) to demonstrate altered levels of betaine from peripheral blood samples of patients with a FESZ.

Ohnishi and colleagues use a variety of state-of-the-art translational approaches to analyze betaine metabolic pathways on both mouse and human experimental models [START_REF] Ohnishi | Investigation of betaine as a novel psychotherapeutic for schizophrenia[END_REF]. They focused on Choline Dehydrogenase (CHDH) enzyme that is involved in betaine biosynthesis. They first generated Chdhknockout mice in order to analyze transcriptomic changes in the frontal cortex that is known to be involved in schizophrenia pathophysiology. They also studied CHDH variant, rs35518479 that was identified as a cis-expression quantitative trait locus (QTL) for CHDH expression using postmortem brains from schizophrenia patients. Because coexistent betaine pathology and elevated carbonyl stress were evident in a subset of schizophrenia cases, they also generated glyoxylase 1 (GLO1)deficient human induced Pluripotent Stem Cell (hiPSC) lines by harnessing the CRISPR-Cas9 system.

They also studied the effects of betaine supplementation in inbred B6N and C3HN mouse strains. This variety of translational approaches were instrumental to showing that betaine metabolic pathway is indeed involved in schizophrenia-related endophenotypes, but also that the CHDH variant, rs35518479, allows genotype-based stratification of schizophrenia patients for betaine efficacy. These two foremost findings open new avenues for therapeutic approaches.

Identification of "a molecular precursor signature" underling FESZ pathophysiology is an important issue for future basic research in this field. The same group led by Dr. T. Yoshikawa already tackles © 2019 published by Elsevier. This manuscript is made available under the CC BY NC user license https://creativecommons.org/licenses/by-nc/4.0/ Version of Record: https://www.sciencedirect.com/science/article/pii/S235239641930461X Manuscript_a3449b898932dcec7d6ad7d3e53be685 this issue using another mouse model based on gestational and early postnatal dietary deprivation of two Polyunsaturated fatty acids (PUFAs)-arachidonic acid (AA) and docosahexaenoic acid (DHA) [START_REF] Maekawa | Polyunsaturated fatty acid deficiency during neurodevelopment in mice models the prodromal state of schizophrenia through epigenetic changes in nuclear receptor genes[END_REF]. This model mimics effects on the brain evidenced in offspring whose mothers experience malnutrition during pregnancy, shown to be a risk factor for schizophrenia. In this article, the authors found downregulation of genes in the prefrontal cortex involved in oligodendrocyte integrity and the gamma-aminobutyric acid (GABA)-ergic system. Regulation of these genes was mediated by the nuclear receptor genes Rxr and Ppar, whose promoters were hypermethylated. Manganese-enhanced MRI that allows detecting activated neurons through calcium channels identified higher signals in the medial prefrontal cortex (mPFC) and the nucleus accumbens shell.

In EBioMedicine, Ohnishi et al. [START_REF] Ohnishi | Investigation of betaine as a novel psychotherapeutic for schizophrenia[END_REF] analyzed transcriptomics of Chdh knockout mice. They first found a deregulation of eIF2 signaling that is known to be central to synaptic plasticity, learning and memory. Furthermore, transcriptomics of Chdh knockout mice frontal cortex demonstrates upregulation of genes enriched for molecular pathways related to the schizophrenia pathophysiology such as glutamate receptor signaling and synaptic long-term depression.

Altogether, these results indicate that FESZ molecular signature involves deregulation of synaptic pathways, as already found from genome-wide association studies of schizophrenia patients [START_REF]Schizophrenia Working Group of the Psychiatric Genomics Consortium. Biological insights from 108 schizophrenia-associated genetic loci[END_REF]. This genome-wide association study included up to 36,989 cases and 113,075 controls and identified 128 independent associations spanning 108 conservatively defined loci that meet genome-wide significance, including loci with genes involved in glutamatergic neurotransmission.

These findings point to the need for further translational research especially using unique mouse molecular engineering toolbox to generate novel approaches. Recent advances on FESZ demonstrated changes in brain structure, in particular in the anterior cingulate cortex and the insula [START_REF] Stefansson | CNVs conferring risk of autism or schizophrenia affect cognition in controls[END_REF], signals from visual [START_REF] Silverstein | Schizophrenia and the eye[END_REF] and auditory pathways [START_REF] Kozono | Auditory Steady State Response; nature and utility as a translational science tool[END_REF]. Both structural changes using MRI approaches and methodologies to explore either visual or auditory pathways are available in mice. Furthermore, Huckins et al. [START_REF] Huckins | Gene expression imputation across multiple brain regions provides insights into schizophrenia risk[END_REF] used transcriptomic imputation approaches that combine eQTL reference panels with large-scale genotype data in order to test associations between disease and gene expression. They demonstrated that genes encoding proteins involved in metabolic pathways are indeed associated with schizophrenia.

The results obtained by Ohnishi et al. need further studies based on larger cohorts of FESZ in order to exclude possible confounding effects between FESZ and treatment, as it should be considered that most patients with FESZ received medication.

In conclusion, the present study underscores the potential benefit of betaine in a subset of schizophrenia and a possible precision medicine strategy by stratifying schizophrenia patients for betaine efficacy. Furthermore, identification of novel biomarkers related to betaine metabolic pathways could promote earlier detection and better treatment options, eventually leading to a better prognosis.

Disclosure

The author declared no conflicts of interest.