A proactive condition-based maintenance strategy with both perfect and imperfect maintenance actions

Reliability Engineering and System Safety 133 (2015) 22–32

Contents lists available at ScienceDirect

Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

A proactive condition-based maintenance strategy with both perfect and imperfect maintenance actions Phuc Do n, Alexandre Voisin, Eric Levrat, Benoit Iung Lorraine University, CRAN, CNRS UMR 7039, Campus Sciences BP 70239, 54506 Vandoeuvre, France

art ic l e i nf o

a b s t r a c t

Article history: Received 7 January 2013 Received in revised form 19 June 2013 Accepted 25 August 2014 Available online 6 September 2014

This paper deals with a proactive condition-based maintenance (CBM) considering both perfect and imperfect maintenance actions for a deteriorating system. Perfect maintenance actions restore completely the system to the ‘as good as new’ state. Their related cost are however often high. The first objective of the paper is to investigate the impacts of imperfect maintenance actions. In fact, both positive and negative impacts are considered. Positive impact means that the imperfect maintenance cost is usually low. Negative impact implies that (i) the imperfect maintenance restores a system to a state between good-as-new and bad-as-old and (ii) each imperfect preventive action may accelerate the speed of the system's deterioration process. The second objective of the paper is to propose an adaptive maintenance policy which can help to select optimally maintenance actions (perfect or imperfect actions), if needed, at each inspection time. Moreover, the time interval between two successive inspection points is determined according to a remaining useful life (RUL) based-inspection policy. To illustrate the use of the proposed maintenance policy, a numerical example finally is introduced. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Imperfect maintenance Condition-based maintenance Remaining useful life Stochastic process Optimization

1. Introduction Maintenance involves preventive and corrective actions carried out to retain a system in or restore it to an operating condition. Optimal maintenance policies aim to provide optimum system reliability/availability and safety performance at lowest possible maintenance costs [23]. In the literature, perfect maintenance actions (or replacement actions) which can restore the system operating condition to as good as new have been considered in various maintenance models. The implementation of “perfect” maintenance policies seems quite simple, however, perfect maintenance actions are often expensive. Imperfect maintenance implying that the system condition after maintenance is somewhere between the condition before maintenance and as good as new has grown recently as a popular issue to researchers as well as industrial applications, see for example [2,12,14,15,17,20]. From a practical point of view, imperfect maintenance can describe a large kinds of realistic maintenance actions [23]. In the imperfect maintenance approach, imperfection can be considered arising from two main causes:


the “bad” realization of a perfect maintenance action due to, for instance, human factors (e.g. stress, lack of skills, lack of attention …), lack of spare parts, lack of repair time …

n

Corresponding author. E-mail address: [email protected] (P. Do).

http://dx.doi.org/10.1016/j.ress.2014.08.011 0951-8320/& 2014 Elsevier Ltd. All rights reserved.


the maintenance policy of decreasing costs such as contractualization which may lead to deal with “lowcost” people, spare parts, logistics …implying for instance. While the first cause does not provide any benefit, the second one may lead to cost benefits. It may however conduct short and longterm deterioration of the maintained system because imperfect maintenance does not make the system return to as good as new state. While short deterioration may be low enough for the system, their accumulations induce long-term deterioration that could be technically or/and economically no-more acceptable. To cope with such a situation, one can image to make a perfect maintenance action at this stage making the system as good as new state, then to take over with imperfect maintenance. In this paper we propose to optimize such maintenance policy mixing perfect and imperfect maintenance actions. Various methods and optimal policies for imperfect maintenance are summarized and discussed in [23,29]. In such maintenance models, preventive maintenance decision is however based on the system age and on the knowledge of the statistical information on the system lifetime. As a consequence, the realistic operating conditions of the system over time are not be taken into account. To face this issue, condition-based maintenance (CBM), for which preventive maintenance decision is based on the observed system condition, has been introduced. Thanks to the rapid development of monitoring equipment which provides accurate information about the system condition over time, CBM

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

Nomenclature Ci Cp Ckp Cc Cd d(t) C t ð:Þ Cð:Þ L K M mð:Þ Ni(t) Np(t)

inspection cost perfect (replacement) preventive maintenance cost cost of the kth imperfect maintenance action corrective (replacement) maintenance cost unavailability cost rate of the system total time passed in failed state in ½0; t the cumulative maintenance cost in ½0; t long-run expected maintenance cost per unit of time failure threshold imperfect maintenance threshold preventive maintenance threshold time interval between two successive inspection points number of inspection in ½0; t number of perfect preventive maintenance in ½0; t

becomes nowadays more and more popular approach in industrial application. Various CBM policies have been proposed and applied for many industrial systems, see for example [9,10,21,27,28]. Starting from the CBM policy, new trend of maintenance strategy has lead to anticipate the failure. Hence, the degradation monitoring is followed by a prognostic step. Among such strategies, one finds CBM þ, PHM (Prognostics and Health Management), proactive maintenance. The use of prognostic is dedicated to the forecast of the remaining useful life (RUL) before the failure occurs. Moreover, it is usually performed during the use of the material/ system in order to adapt the maintenance policy. Indeed, the prognostic algorithms are fed with monitored data in order to predict the RUL in the current condition. In the prognostic review [11,26], one can classify the prognostic approaches into 3 groups:


model based: in this class the prognostic approaches are based



on a model representing the physics of the degradation process built with specific experiments; data based: this class includes approaches using monitored data in order to construct the model; statistics/experience based: this class corresponds to reliability or stochastic models based on feedback data (i.e. mainly failure data).

While the first two approaches require monitored data and are used on-line, the last one may be used off-line. In such a way, using feedback data one can use experience based prognostic model in order to optimize a priori maintenance policy. In the proposed CBM maintenance policy, RUL prognosis is used in order to plan the next inspection time at which maintenance decision will be taken. It has been recently shown in [18,22,24] that combining CBM and imperfect maintenance is well suited. Indeed, according to the observed condition of the system, an optimal maintenance action represented by an optimal intervention gain is preventively carried out. However, in such maintenance policies, only imperfect preventive or imperfect repair actions are considered and the system is assumed to be imperfectly maintained an infinite number of times. From a practical point of view, this assumption may not always be relevant since, in a variety of engineering and service applications, systems can be maintained only a limited number of times due to technical or economical reasons [13]. Furthermore, as mentioned in [22], each imperfect maintenance action may make the system more susceptible to future deterioration. To this end, a fixed number of

Nip(t) Nc(t) Q Ti Xt Zk uð:Þ vk

α0 ; β

αk η γ

23

number of imperfect preventive maintenance in ½0; t number of corrective maintenance in ½0; t failure probability between two inspection times ith inspection time system deterioration level at time t kth intervention gain degradation improvement factor mean deterioration speed after the kth maintenance action scale and shape parameters of the deterioration process when the system is as good as new scale parameter of deterioration process after the kth imperfect maintenance action a non-negative real number non-negative real number and represents the impact of imperfect maintenance actions on the deterioration speed of the system

allowable imperfect maintenance actions is introduced in maintenance models in [13,5] and considered as a decision parameter. However, the value of this decision parameter is arbitrary chosen and they do not describe how the imperfect repair actions affect the deterioration evolution of the system. As a consequence, two important issues arise. The first one concerns the investigation on the impacts of imperfect maintenance actions to the deterioration of the system. The second issue relies on how to determine the optimal number of imperfect maintenance actions for each life cycle of the system. To overcome these issues, the aim of this paper is to propose a proactive condition-based maintenance policy with both perfect and imperfect maintenance actions for a deteriorating system. The first original contribution of the paper concerns the investigations of imperfect maintenance actions. Both positive and negative impacts are considered. Positive impact means that it can reduce the deterioration level of the system with reduced maintenance cost. Negative impact implies that (i) the deterioration level of the system after imperfect maintenance may not be reset to zero and (ii) each imperfect preventive action may accelerate the speed of the system's deterioration process. The second original contribution of the paper is to propose an adaptive maintenance policy. Different maintenance rules which can help to select optimally maintenance actions at each inspection time are proposed. Based on the proposed maintenance policy, the optimal number of imperfect maintenance action for each cycle of the system is determined. Moreover, in CBM practice, inspections are usually performed at regular intervals. However, it may not be always profitable to inspect the system at regular intervals of time, especially when the inspection procedure is costly. The present paper proposes to use an aperiodic inspection policy which is based on the prognosis of the residual useful life (RUL) of the system, see [3,8,30]. The rest of the paper is organized as follows. Section 2 is devoted to the descriptions of the characteristics of the system to be maintained and the related assumptions. Deterioration modeling is also described. Section 3 focuses on the investigations of the impacts of imperfect maintenance actions. Their related cost model are also described and discussed. An adaptive maintenance policy is described in Section 4. To illustrate the proposed maintenance policy, some numerical values of the considered system and its related context are introduced in Section 5. Some numerical results are in addition discussed here. Finally, the last section presents the conclusions drawn from this work.

24

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

2. System description and assumptions 2.1. General assumptions Consider a system which can be simplified by a single component system (e.g. only the most important component is considered). The condition (deterioration level) of the system at time t is assumed to be summarized by an observable random scalar variable Xt. In the absence of repair or replacement actions, the evolution of the system deterioration is assumed to be strictly increasing (e.g. the evolution of a crack length or of the number of defect products). The process ðX t Þt Z 0 can be then an increasing stochastic process. In addition, the following assumptions are considered.


The initial state X0 is 0.
The system is failed if its deterioration level is greater than a




level L. The threshold L can be seen as a deterioration level which must not be exceeded for economical or security reasons. The system failure is self-announcing.

of an infinite number of small shocks. Following this spirit, it is assumed that the deterioration of the system between the kth and the (kþ 1)th maintenance actions evolves like a Gamma stochastic process ðX~ t Þt Z 0 , with the following characteristics:
X~ 0 ¼ X k , (Xk represents the deterioration level of the system after the kth maintenance action).
ðX~ t Þt Z 0 has independent increments.
For all 0 r l o t, the random increment X~ t  X~ l follows a Gamma probability density (pdf) with shape parameter αk ðt lÞ and scale parameter β: f αk ðt  lÞ;β ðxÞ ¼

1

Γ ðαk ðt  lÞÞ

βαk ðt  lÞ xαk ðt  lÞ  1 e  βx I fx Z 0g ;

where: ○ I fx Z 0g is an indicator function I fx Z 0g ¼ 1 if x Z 0, I fx Z 0g ¼ 0 otherwise. ○ αk ¼ vk =β with vk being the mean deterioration speed of the system between the kth and the (kþ1)th maintenance actions.

The system degradation behavior and corresponding states are illustrated in Fig. 1. It is assumed that continuous monitorings are impossible (e.g. monitoring equipments are not integrated in the system due to whatever reason). Inspections are needed and only discrete inspections are possible. Each inspection incurs a cost Ci. To avoid failure occurrence of the system, preventive maintenance is considered. It is assumed that both imperfect and perfect preventive maintenance actions for the system are possible. This assumption may no relevant for some kinds of systems (e.g. critical systems) where imperfect maintenance actions may be impossible due to security or technology raisons.

Fig. 2 illustrates a possible degradation paths and the corresponding failure distribution at time 0 when α0 ¼ 1 and β ¼ 1. After a corrective or perfect preventive maintenance action, the system becomes as good as new (the deterioration level is reset to 0 and the deterioration behavior evolves with time according to the nominal speed v0 ¼ α0 =β). Imperfect maintenance actions can reduce the system's deterioration level with reduced maintenance costs. However, as mentioned in [13,22], imperfect maintenance actions may affect the evolution of the system's deterioration process. The impacts of imperfect maintenance actions will be described in the next section.

2.2. Deterioration modeling

3. Imperfect maintenance actions and related costs

Gamma processes have been widely used to describe the degradation of systems [5,10,28]. A characteristic of this process is that it is strictly monotone increasing which is the behavior observed in most physical deterioration processes. Moreover, its paths are discontinuous and it can be thought as the accumulation

3.1. Impact of imperfect actions on the deterioration level In this paper, we consider that the imperfect preventive maintenance actions lead the system to a better state for which the degradation level is lower or equal to the current deterioration

Fig. 1. Illustration of the system degradation evolution and its state.

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

25

Fig. 2. Illustration of possible degradation paths and initial failure distribution.

Fig. 3. Illustration of deterioration evolution and impacts of imperfect maintenance.

level of the system. It is shown in the literature that maintenance gains, defined as the reduction quantity on the deterioration level of the system due to an imperfect maintenance action, can be random, see for instance [2,5,18]. In this way, if the kth imperfect maintenance action is performed at inspection time Ti, the intervention gain is then assumed to be described by a continuous random variable Zk. Zk is bounded, i.e., 0 r Z k rX T i where X T i is the deterioration level of the system at Ti. In fact, it is shown in [5] that Zk can be distributed according to a truncated normal distribution with density:

g μ;σ ;a;b ðxÞ ¼


 1 ϕ xμ  σ  σ
I ½a;b ðxÞ; Φ b σ μ  Φ a σ μ

ð1Þ

where:


I ½a;b ðxÞ ¼ 1pffiffiffiffiffiffi if a r x r b and I ½a;b ðxÞ ¼ 0 otherwise.
ϕðξÞ ¼ 1= 2π expð  12ξ2 Þ is the probability density function of the standard normal distribution and ΦðÞ is its cumulative distribution function.


μ ¼ X T =2 and σ ¼ X T =6.
a ¼ μ  3σ ¼ 0 and b ¼ μ þ 3σ ¼ X T . i

i

i

According to this distribution, it is clear that a rZ T i r b, i.e. 0 r Z k r X T i is satisfied. The average of intervention gains is EðZ k Þ ¼ μ and the variance is VARðZ k Þ ¼ 0:973σ 2 , see [25]. It is assumed that the intervention gain Zk can be measurable (e.g. crack length or vibration of the system after maintenance can be measurable). Thanks to the imperfect

26

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

preventive action, the deterioration level of the system after maintenance is set to X k ¼ X T i Z k . The illustration of the system deterioration evolution and random maintenance gain are shown in Fig. 3.

3.2. Impact of imperfect actions on the deterioration speed To model the impact of imperfect actions on the evolution deterioration of the system, it is assumed in this work that each imperfect preventive action affects the speed of the system deterioration process. This can be found in a variety of business sectors, e.g. welding can reduce crack length but it may destroy some physical behaviors of materials; removing several parts of the system for maintenance actions may accelerate the deterioration evolution of other parts; spare parts may be reusable components or low quality components, as a consequence, after maintenance the deterioration level of the system can be reduced however the deterioration speed may be increased. The impact of an imperfect maintenance action on the system deterioration speed can be described by non-negative continuous random variable ϵk which follows an exponential distribution with density probability:

In general, each maintenance action incurs a cost and an imperfect maintenance action often incurs a reduced maintenance cost, namely imperfect maintenance cost. This cost may be independent of intervention gain Zk and bounded by perfect maintenance cost, see for instance [2,14,18]. From a practical point of view, in most cases, the quality of the maintenance action increases with the level of resources allocated to it, and hence with its cost, see [16,17,19]. To model imperfect maintenance actions in the context of deteriorating systems, the degradation improvement factor, defined as the ratio of the improvement gain divided by the deterioration level of the system before maintenance, has been recently introduced, see [5] as an efficient indicator. Based on the improvement factor, imperfect maintenance costs can be evaluated and considered as a function of the improvement factor. In that sense, it is assumed in this work that when the kth imperfect preventive action is performed at inspection time Ti, one has to pay a maintenance cost which is defined as C kp ¼ C 0p :uðT i Þη ;

ð3Þ

where:

hðxÞ ¼ γ e  γ x I fx Z 0g ; where γ is a non-negative real number. The mean value of ϵk is E½ϵk  ¼ γ . By this modeling, if the kth maintenance action is a corrective or perfect preventive maintenance, the mean deterioration speed of the system after maintenance is reset to vk ¼ v0 ¼ α0 =β. If the kth maintenance action is an imperfect preventive one, the mean deterioration speed of the system after maintenance is set to v k ¼ v k  1 þ ϵk :

3.3. Imperfect preventive maintenance cost

ð2Þ

It is assumed that ϵk is measurable. An example of increasing of the deterioration speed due to an imperfect maintenance action is illustrated in Fig. 3. A case study on the sensitivity to the effect of imperfect maintenance actions will be discussed in Section 5.


uðT i Þ ¼ Z k =X T is the degradation improvement factor.
C0p is imperfect preventive cost incurred when the deterioration i




level of the system is reduced to 0 with imperfect maintenance action. This cost is usually lower than a perfect preventive cost (C 0p r C p ).η is a non-negative real number.

According to this cost model, different kinds of maintenance cost function can be found depending on the value of η. More precisely:


When η ¼ 0, imperfect maintenance cost is constant (C kp ¼ C 0p Þ.
When 0 o η o 1, imperfect maintenance cost Ckp is a concave function: the maintenance cost increases more than the improvement gain when performing the maintenance.

Fig. 4. Illustration of imperfect maintenance cost function.

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32


When η ¼ 1, Ckp is a linear function which implies that the



If L 4 X T Z M, the system is still functioning, however its i

maintenance cost is proportional to the improvement level gain. When η 4 1, Ckp is a convex function: the maintenance cost increases less than the improvement gain.

Fig. 4 illustrates these three different shapes of the imperfect maintenance cost function. A case study of the proposed imperfect maintenance policy with different kinds of imperfect maintenance cost functions will be presented in Section 5.

4. Maintenance policy In the framework of CBM optimization, a maintenance policy relies essentially on two main decisions: when to take (preventively/correctively) maintenance actions and when to inspect. To this end, the deterioration level of the system can be used to make the decision on the inspection time and on the maintenance action to be performed [10,28,4]. The maintenance decision is herein based on both the system deterioration level at inspection time and the future evolution of the system's deterioration process. More precisely, according to the degradation level X T i at inspection time Ti, the maintenance decision is the following:


If X T oM ðM r LÞ, the system is in a working state, no maini

tenance action is performed. M is called the preventive maintenance threshold and it is a decision variable to be optimized.

27




deterioration level is considered as ”warning”. Hence, a preventive maintenance action is immediately carried out. Without loss of generality, it is assumed that this preventive maintenance action is the kth preventive maintenance action from the last perfect maintenance of the system. If k¼ K (K is called the imperfect preventive threshold and it is a decision variable to be optimized), the kth preventive maintenance action is a perfect one. Thanks to this preventive perfect maintenance action, the deterioration level of the system after maintenance X T i is reset to zero. Contrarily, if k oK the kth preventive maintenance action is an imperfect one. This imperfect preventive maintenance action can lead the system to a better state for which the degradation level is lower or equal to the current deterioration level of the system. The impacts of imperfect preventive maintenance actions and their related cost have been described in Section 3. if X T i Z L , the system is failed, then a corrective replacement action is performed and a cost Cc is incurred. An additional cost is incurred by the time dj elapsed in the failed state at a unavailability cost rate Cd which may correspond to, for example, production loss per unit of time. After a corrective maintenance action, the system is considered as good as new (the deterioration level of the system after maintenance is reset to zero).

It is assumed finally that maintenance durations are neglected and all the necessary maintenance resources to execute preventive or corrective maintenance actions are always available. It must be noticed that this assumption may be not relevant for some realistic cases, e.g. logistic support need to be prepared; maintenance

Fig. 5. Decision process of the proposed maintenance policy.

28

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

crews are not always available. However, taking into account these kinds of activities into the proposed maintenance policy, the problem becomes much more complex and it should be investigated by other studies. The decision process of the maintenance policy is illustrated in Fig. 5. As shown in the maintenance decision process, the inspections are indispensable in the proposed maintenance policy. An inspection policy is described in the next section.

with, mðX T i ; Q Þ ¼ fΔT : PðX T i þ ΔT Z LjX T i Þ ¼ Q g; where: PðX T i þ ΔT Z LjX T i Þ ¼ PðX T i þ ΔT  X T i ZL X T i Þ Z

1

¼ L  XTi

Z ¼ 1 0

4.1. RUL based inspection Different inspection policies, which aim to optimize the time interval between two successive inspection points, have been introduced in the literature. In fact, the time interval between two successive inspection points can be fixed regardless of the degradation level, e.g. [24], or aperiodic and deterioratingdependant via an inspection scheduling linear [10], or non-linear [1] function with respect to the deterioration level. Residual Useful Life (RUL) based inspection has been recently introduced, see [3,8,30]. The latter seems very promising especially in the context of condition-based maintenance. Remaining useful life (RUL) is defined as the duration left for a system before it fails. RUL can be considered also as a residual duration for which the system fails with a given probability. The later one is used in this work. The main idea of the RUL based inspection is that the next inspection time is chosen such that the probability of the failure of the system before the next inspection remains lower than a limit Q (0 oQ r 1). Q is a decision variable to be optimized. If we let Ti denote the time at which the system is inspected, and the corresponding degradation level of the system is X T i (it is the deterioration level of the system after maintenance if a maintenance action is executed at time Ti), the next inspection time is then determined by T i þ 1 ¼ T i þ mðX T i ; Q Þ;

ð4Þ

ð5Þ

f αk ΔT;β ðxÞ dx L  XTi

f αk ΔT;β ðxÞ dx:

ð6Þ

It is clear that mðX T i ; Q Þ depends on the current degradation level of the system, the failure threshold L and the parameter Q [8]. The illustration of the time interval between two successive inspection points is shown in Fig. 2. The integration of mðX T i ; Q Þ in the maintenance decision process is illustrated in Fig. 6. Finally, the inspections are assumed to be instantaneous, perfect and non-destructive. When an inspection is performed, a cost Ci is incurred. According to this inspection policy, the reliability of the system between two inspection times interval remains higher or equal to (1  Q ). This means that the proposed maintenance policy can provide an optimal maintenance planning with a given reliability level. From a practical point of view, this result seems to be very interesting since in many industrial systems, the reliability of the system may be an important constraint due to technical and/or economical reasons, see for example [7]. A case study will be discussed in Section 5.

4.2. Optimization of the maintenance policy To evaluate the performance of the maintenance policy, the long-run expected maintenance cost rate including the unavailability cost is used herein as the main criterion in order to final the optimal decision parameters M, K and Q.

Fig. 6. Illustration of degradation behavior and the proposed maintenance policy impact.

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

According to the proposed maintenance policy, the cumulative maintenance cost at time t is Nip ðtÞ

t

C ðM; K; Q Þ ¼ C i :Ni ðtÞ þ ∑

k¼1

C kp þN p ðtÞC p þ

Nc ðtÞ

∑ C c þ C d :dðtÞ;

ð7Þ

j¼1

where: N i ðtÞ, N p ðtÞ, N ip ðtÞ, N c ðtÞ are respectively the number of inspections, of perfect preventive maintenance, of imperfect maintenance and of corrective replacement in ½0; t; d(t) is the total time passed in a failed state in ½0; t. By using the renewal theory [25], the long run expected maintenance cost per unit of time is CðM; K; Q Þ ¼ lim

t-1

C t ðM; K; Q Þ E½C H ðM; K; Q Þ ¼ ; t E½H

ð8Þ

where H is the first replacement date (H represents the life cycle length of the system) and E½: represents mathematical expectation. Stochastic Monte Carlo simulation is used to evaluate this cost criterion. To do so, for each value of decision parameters (M; K; Q ), the corresponding maintenance cost rate is calculated. To ensure that the convergence of maintenance cost rate is reached, a large number of simulation realizations (each realization simulates one life cycle of the system) must be done. By varying different values of (M; K; Q ), the minimum maintenance cost rate can be identified. The optimal values of the decision parameters (M; K; Q ) are obtained when the minimum maintenance cost rate is reached, i.e, CðM n ; K n ; Q n Þ ¼ min fCðM; K; Q Þ; 0 r M o L; 0 r K; 0 o Q o1g: M;K;Q

ð9Þ

4.3. Flexibility of the proposed maintenance policy According the proposed maintenance policy, the interest of imperfect maintenance policy is represented by Kn. When K n ¼ 0, no imperfect action is considered, the proposed maintenance policy becomes a perfect policy whose performance is investigated and proved in [4,10]. When 0 o K n o 1, the proposed maintenance policy leads to a hybrid policy in which both perfect and imperfect maintenance actions are considered. It must be noted that the higher Kn is, the more the interests of imperfect actions are. For the last case K n ¼ 1, even it may be only a theoretical case where all preventive maintenance actions are imperfect. The proposed maintenance policy corresponds to an imperfect maintenance one. The flexibility of the proposed maintenance policy is illustrated in Fig. 7.

29

5. Numerical example The purpose of this section is to show how the proposed maintenance policy can be used in maintenance optimization through an example whose characteristics are described in Section 2. Consider a deteriorating system in which its degradation behavior, when no maintenance is carried out, is assumed to be described by a Gamma process with scale parameter α0 ¼ 1 and shape parameter β ¼ 1. If the degradation of the system exceeds the failure threshold L¼20, the system is failed. Both corrective and perfect preventive maintenances can restore completely the system to the ‘as good as new’ state. Besides, the deterioration level of the system can be improved by imperfect maintenance actions which may however affect the deterioration speed, see again Section 2.2. Table 1 reports the data related to inspection, maintenance costs, unavailability cost rate (all costs are given in arbitrary units) and the impact of imperfect maintenance actions on the deterioration speed. To evaluate the mean maintenance cost per unit of time, a very large number of simulation realizations are done. In order to find the optimal decision parameters (M; K; Q ), the average of maintenance cost per unit of time CðM; K; Q Þ is evaluated with different values of M (0 r M o L), K (K Z 0) and Q (0 o Q o 1) by using Eq. (8). The optimum values of the decision parameters are M n ¼ 14, K n ¼ 4 and Q n ¼ 0:10 with the minimum maintenance cost rate CðM n ; K n ; Q n Þ ¼ 5:15. To compare with a perfect maintenance policy (only the maintenance cost criterion is herein used), K is firstly set to zeros (as mentioned in Section 4.3, when K ¼ 0, the proposed maintenance policy corresponds to a perfect RUL based maintenance policy which seems to be an efficient policy in the framework of perfect condition-based maintenance [4,10]). The average of maintenance cost per unit of time is then evaluated with different values of M and Q. The minimum average maintenance cost rate is finally found at 6.23 which is higher (about 21%) than the result obtained by the proposed policy. Moreover, a sensitivity analysis of the proposed maintenance policy with respect to the number of imperfect actions within a life cycle is studied and sketched in Fig. 8. The results show that when K o 4, the maintenance cost rate increases quickly if K is close to 0 and when K 44 the maintenance cost rate increases slowly with respect to the increasing of K. According to these results, it is clear that, with the considered system whose data is Table 1 Data of costs and impact of imperfect actions. Ci

Cc

Cp

C0p

Cd

η

γ

10

100

90

70

20

3

0.2

Fig. 7. Proposed maintenance policy and its related policies.

30

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

Fig. 8. Mean maintenance cost rate as a function of K.

Table 2 Optimal maintenance policy with a given η. η

0 0.4 1 2 3 5

Table 3 Optimal maintenance policy with a given γ. CðM n ; K n ; Q n Þ

Optimal decision parameters Mn

Kn

Qn

16 16 16 14 14 12

0 0 0 2 4 6

0.15 0.15 0.15 0.13 0.10 0.08

6.23 6.23 6.23 5.72 5.15 4.44

reported in Table 1, imperfect maintenance actions seem to be more appropriate than perfect ones. 5.1. Sensitivity analysis to the imperfect maintenance cost The performance of imperfect maintenance actions may depend on their related cost which are, in this paper, characterized by η(see again Eq. (3)). Table 2 reports the optimum values of M, K and Q and the minimum value of CðM; K; Q Þ for different values of η. The results show that when η r 1, the maintenance cost rate remains unchanged, the optimal maintenance policies correspond to a perfect maintenance policy (K ¼0). However, the maintenance cost rate decreases dramatically when η 41. This means that, in this case study, the imperfect maintenance cost has a significant influence on the interest of the imperfect maintenance actions. 5.2. Sensitivity analysis to the impact of imperfect maintenance on the deterioration speed To analyze the impact of imperfect maintenance actions, different values of γ are considered. For each value of γ, a maintenance policy characterized by the decision parameters (M; K; Q ) is optimally found using Eq. (9) and the obtained results are reported in Table 3.

γ

0.05 0.1 0.2 0.5 1 2 3

CðM n ; K n ; Q n Þ

Optimal decision parameters Mn

Kn

Qn

12 13 14 15 15 16 16

13 5 4 2 1 0 0

0.06 0.07 0.10 0.11 0.12 0.15 0.15

4.31 4.78 5.15 5.69 5.99 6.23 6.23

Table 4 Optimal policy with a given reliability level. R

0.99 0.97 0.95 0.93 0.90 0.85

CðM n ; K n ; Q n Þ

Optimal decision parameters Mn

Kn

Qn

12 13 13 14 14 14

3 3 3 3 4 4

0.01 0.03 0.05 0.07 0.10 0.10

6.11 5.58 5.35 5.26 5.15 5.15

The results show that when the impact of imperfect actions on the deterioration speed of the system is small, the maintenance cost is relatively low. Oppositely, when the impact of imperfect actions on the deterioration speed is large (e.g., γ Z 2), the maintenance cost is high and the proposed maintenance policy becomes a perfect one (K ¼ 0). This can be explained by the fact that imperfect maintenance actions are cheaper than perfect ones however they are indirectly penalized by their negative influence on the deterioration speed of the system. As a consequence, when their effect on the system's deterioration process is small, they become more appropriate and when this effect is large, perfect maintenance actions seem to be a better choice.

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

31

Fig. 9. Maintenance cost rate as a function of reliability level.

5.3. Optimal maintenance policy for a given reliability level It is assumed now that the system is required to operate with a given reliability level R. The optimal maintenance policy obtained above may not be an optimal policy in this context. We are interested herein to find an optimal maintenance policy which can satisfy this demand. To this end, according to the proposed maintenance policy, we simply take the decision parameter 0 o Q r 1  R. This means that the optimal policy can be found by changing the values of the decision parameters M (0 rM o L), K (K Z 0) and Q (0 o Q r 1  R). Table 4 reports the obtained results with different values of the system reliability. It is not surprising that: the higher level the reliability level has, the larger cost the maintenance incurs. The maintenance cost rate as a function of reliability level is shown in Fig. 9. To compare with the perfect maintenance policy given a reliability level R, K is set to zero. The optimal perfect policy can be found by changing the decision parameters Q (0 o Q r 1  R) and M (0 r M o L). The results represented in Fig. 9 show that given a reliability level, the imperfect maintenance policy costs less than the perfect one. This means that under a limited maintenance cost, the imperfect policy leads to a higher reliability level than the perfect policy does. Moreover, it must be noted that the lowest maintenance cost rate provided by the perfect policy is found when the reliability level of the system is 0.85. This lowest cost is still however higher than the cost (6.11) obtained by the imperfect maintenance policy given a reliability level of 0.99, see Fig. 9.

6. Conclusions In this work, a proactive condition-based maintenance (CBM) policy with both perfect and imperfect maintenances for a deteriorating system is proposed. Imperfect maintenance actions characterized by random intervention gains are studied and discussed with different types of their related cost which may be a concave, linear or convex function with respect to intervention gain. The impact of imperfect actions on the deterioration speed of the system is also investigated. Moreover, a novel maintenance policy is proposed. Thanks to the residual useful life (RUL) based inspection policy, the proposed maintenance policy can provide an optimal maintenance planning with a given reliability level. The

proposed imperfect maintenance policy may optimally become a perfect one [10,4] for several cases, e.g., when the imperfect cost is high or/and imperfect actions largely affect the deterioration speed of the system. Finally, the performance of the proposed policy is illustrated and discussed through some numerical values of a deteriorating system. Different sensitivity analysis is investigated to show the advantages of the proposed maintenance policy. It must be noted finally that, in this paper, Gamma stochastic process is used to describe the deterioration of the system to be maintained however, the proposed maintenance policy can be applied for other kinds of systems with different deterioration behaviors (e.g., deteriorations modeled by Wiener, Markov processes, …). This paper is the development of our research in the framework of imperfect condition-based maintenance models presented in part in [6]. Our future research work will focus on the integration of prognostic to estimate the RUL in the proposed CBM model. Logistic support will be studied and integrated in our proposed policy. Furthermore, we hope to develop the proposed CBM policy for multi-component system. References [1] Barker C, Newby M. Optimal non-periodic inspection for a multivariate degradation model. Reliab Eng Syst Saf 2009;94:33–43. [2] Castro IT. A model of imperfect preventive maintenance with dependent failure modes. Eur J Oper Res 2009;196(1):217–24. [3] Cui L, Xie M, Loh H-T. Inspection schemes for general systems. IIE Trans 2004;39(9):817–25. [4] Do Van P, Berenguer C. Condition based maintenance model for a production deteriorating system. In: Conference on control and fault-tolerant systems (SysTol'10). 6–8 September 2010. Nice, France: IEEE; 2010. [5] Do Van P, Berenguer C. Condition-based maintenance with imperfect preventive repairs for a deteriorating production system. Reliab Qual Eng Int 2012;28(6):624–33. [6] Do Van P, Voisin A, Levrat E, Iung B. Condition-based maintenance with both perfect and imperfect maintenance actions. In: Annual conference of the prognostics and health management society 2012, 23–27 September, 2012. Minnesota, USA; 2012. p. 256–64. ISBN: 978-1-936263-05-9. [7] Do Van P, Vu HC, Barros A, Berenguer C. Grouping maintenance strategy with availability constraint under limited repairmen. In: 8th IFAC international symposium on fault detection, supervision and safety for technical processes. SAFEPROCESS-2012, 29–31 August 2012. Mexico City, Mexico; 2012. [8] Gebraeel N, Lawley M, Li R, Ryan J. Residual-life distributions from components degradation signals: a Bayesian approach. IIE Trans 2005;37:543–57. [9] Ghasemi A, Yahcout S, Ouali M. Optimal condition based maintenance with imperfect information and the proportional hazards model. Int J Prod Res 2007;4:989–1012.

32

P. Do et al. / Reliability Engineering and System Safety 133 (2015) 22–32

[10] Grall A, Dieulle L, Bérenguer C, Roussignol M. Continuous-time predictivemaintenance scheduling for a deteriorating system. IEEE Trans Reliab 2002;51: 141–50. [11] Jardine A, Lin D, Banjevic D. A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mech Syst Signal Process 2006;20:1483–510. [12] Kijima M, Morimura H, Suzuki Y. Periodical replacement problem without assuming minimal repair. Eur J Oper Res 1988;37(2):194–203. [13] Kurt M, Kharoufeh J. Optimally maintaining a Markovian deteriorating system with limited imperfect repairs. Eur J Oper Res 2010;205:368–80. [14] Labeau P-E, Segovia M-C. Effective age models for imperfect maintenance. J Risk Reliab 2010;225:117–30. [15] Levitin G, Lisnianski A. Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203. [16] Lie CH, Chun Y. An algorithm for preventive maintenance policy. IEEE Trans Reliab 1986;35(1):71–5. [17] Liu Y, Huang H. Optimal selective maintenance strategy for multi-state systems under imperfect maintenance. IEEE Trans Reliab 2010;59(2):356–67. [18] Meier-Hirmer C, Riboulet G, Sourget F, Roussignol M. Maintenance optimisation for a system with a gamma deterioration process and intervention delay: application to track maintenance. J Risk Reliab 2008;223:189–98. [19] Mettas A. Reliability allocation and optimization for complex systems. In: IEEE Proceedings of the annual reliability and maintainability symposium; 2000. p. 216–21.

[20] Nakagawa T, Yasui K. Optimal policies for a system with imperfect maintenance. IEEE Trans Reliab 1987;R-36(5):631–3. [21] Neves ML, Santiago L, Maia C. A condition-based maintenance policy and input parameters estimation for deteriorating systems under periodic inspection. Comput Ind Eng 2011;61:503–11. [22] Nicolai RP, Frenk J, Dekker R. Modelling and optimizing imperfect maintenance of coatings on steel structures. Struct Saf 2009;31:234–44. [23] Pham H, Wang H. Imperfect maintenance. Eur J Oper Res 1996;94:425–38. [24] Ponchet A, Fouladirad M, Grall A. Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements. Proc Inst Mech Eng O: J Risk Reliab 2011;225(2):105–16. [25] Ross S. Stochastic processes. Wiley series in probability and statistics. New york: John Wiley & Sons, Inc.; 1996. [26] Si X, Wang W, Hu C, Zhou D. Remaining useful life estimation—a review on the statistical data driven approaches. Eur J Oper Res 2010;213:1–14. [27] Tan L, Cheng Z, Guo B, Gong S. Condition-based maintenance policy for gamma deteriorating systems. J Syst Eng Electron 2010;21:57–61. [28] van Noortwijk J. A survey of the application of Gamma processes in maintenance. Reliab Eng Syst Saf 2009;94:2–21. [29] Wu S, Zuo JM. Linear and nonlinear preventive maintenance models. IEEE Trans Reliab 2010;59(1):242–9. [30] Yang Y, Klutke G-A. A distribution-free lower bound for availability of quantile-based inspection schemes. IEEE Trans Reliab 2001;50:419–21.