CHAPTER 1. Stochastic modelling of the effect of preventive and corrective. maintenance on repairable systems reliability

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1 CHAPTER 1 Stochastic modelling of the effect of preventive and corrective maintenance on repairable systems reliability Laurent Doyen and Olivier Gaudoin Institut National Polytechnique de Grenoble Laboratoire LMC BP Grenoble Cedex 9 France Laurent.Doyen@imag.fr, Olivier.Gaudoin@imag.fr The aim of this paper is to propose a general model of the joint effect of corrective and preventive maintenance on repairable systems. The model is defined by the failure intensity of a stochastic point process. It takes into account the possibility of dependent maintenance times with different effects. The modelling is based on a competing risks approach. The likelihood function is derived, so parameter estimation and assessment of maintenance efficiency are possible. 1. Introduction All important industrial systems are subjected to corrective and preventive maintenance actions which are supposed to extend their functionning life. Corrective maintenance (CM), also called repair, is carried out after a failure and intends to put the system into a state in which it can perform its function again. Preventive maintenance (PM) is carried out when the system is operating and intends to slow down the wear process and reduce the frequency of occurrence of system failures. The assessment of the efficiency of these maintenance actions is of great practical interest, but it has been seldom studied. Several models of maintenance effect have been proposed (see for example a review in Pham and Wang 12 ). Most of them, known as repair models, consider only the effect of CM. Others consider only the effect of PM. In fact, the same kinds of assumptions are done in both cases, so the same models can be considered for the effect of both kinds of maintenances. Usually, these models are defined by the conditional distributions of successive interfailure times. For comparison purpose and in view of a statistical 1

2 2 L. Doyen and O. Gaudoin analysis, we think, as in Lindqvist 10,thatitismoreconvenient to define a model by the failure intensity of a stochastic point process. In this case, maintenance effect is characterized by a change on the failure intensity. Section 2 presents this approach for the case where only CM are considered. Some usual repair models are mentioned as particular cases of the general modelling. Section 3 presents some models taking into account both CM and deterministic planned PM. In section 4, we propose a general modelling mixing CM and random PM, based on a competing risk approach. Within this modelling, the distribution of the time to next failure and likelihood function are derived. 2. Corrective maintenance models 2.1. Stochastic modelling of the failure process Let ft i gi 1 be the successive failure times of a repairable system not subjected to PM actions, starting from T 0 =0.LetX i = T i T i 1 ;i 1, be the successive interfailure times and N t be the number of failures observed up to time t. The repair times are assumed to be negligible or not taken into account, so failure times are equal to CM times. When the occurence of a failure at anytime is assumed to depend only on the own past of the failure process, this process is a self excited point process, and it is characterized by its failure intensity, defined as: 8t 0; N 1 t = lim dt!0 dt P (N t+dt N t =1jFt(N)) where Ft(N) =ff(fn s g 0»s»t )istheinternal history of the failure process at time t. But external variables can also influence failures. If E = fe s gs 0 denotes the process of external variables, the failure intensity relativetoe is defined as: 8t 0; N 1 t (E) = lim dt!0 dt P (N t+dt N t =1jFt(N; E)) where Ft(N; E) =ff(fn s ;E s g 0»s»t ) is here the joint history of the failure process and the external process. In the case of a self excited point process, the failure intensity completely characterizes the failure process. Otherwise, informations about the joint distribution of processes N and E are needed Basic models The basic assumptions on maintenance efficiency are known as minimal repair or As Bad As Old (ABAO) and perfect repair or As Good As New

3 Stochastic modelling of the effect of PM and CM on repairable systems reliability 3 (AGAN). In the ABAO case, each maintenance restores the system in the state it was before failure. The corresponding random processes are the Non Homogeneous Poisson Processes (NHPP). These processes are such that the failure intensity is only a function of time: N t = (t) In the AGAN case, each maintenance perfectly repairs the system and restores it as if it were new. The corresponding random processes are the Renewal Processes (RP). These processes are such that the failure intensity is defined as: N t = (t T Nt ) Obviously, reality is between these two extreme cases: standard maintenance reduces failure intensity but does not restore the system as good as new. This is sometimes known as better-than-minimal repair The Dorado-Hollander-Sethuraman model Many basic repair models can be considered as particular cases of the recent model proposed by Dorado, Hollander and Sethuraman (DHS). 6 This model assumes that there exists two sequences fa i gi 1 and f i gi 1 called respectively the effective ages and life supplements, such that A 1 =0; 1 = 1 and the conditional distributions of the interfailure times are given by: P (X i >xja i ; i ;X 1 ; :::; X i 1 )= F ( i x + A i ) F (A i ) where F is the survival function of the first failure time X 1. This means that after the i th CM, the system behaves as a new one having survived until A i, and its wear-out speed is weighted by a factor i. Denoting by the failure rate of X 1, the failure intensity of the DHS model is : N t (A; ) = Nt+1 (A Nt+1 + Nt+1[t T Nt ]) By giving particular values to A i and i for i 2, several CM models appear to be members of the DHS family. When i = 1 for all i, we obtain the virtual age models proposed by Kijima 8. NHPP and RP belong to this family of models for respectively A i = T i 1 and A i =0. Kijima defined two classes of models, where maintenance effect is expressed by iid random variables D j over [0; 1]. Type I models are such that

4 4 L. Doyen and O. Gaudoin A i = P i 1 j=1 D jx j. When the D j 's are deterministic and constant equal to 1 ρ, the failure intensity is simply : N t = (t ρt Nt ) The model is the Kijima Morimura and Suzuki 9 model, which happens to be the same as the Malik 11 model known as the proportional age reduction (PAR) model. The parameter ρ is known as the improvement factor. Type II models are such that A i = P i 1 j=1 Q i 1 k=j D kx j. When the distribution of the D j 's is Bernoulli, the model is the Brown and Proschan 3 model. When the D j 's are deterministic and constant equal to 1 ρ, the model happens to be the same as the Brown, Mahoney and Sivazlian 2 model. Kijima models with D j =1 ρ can be understood as particular cases of the Arithmetic Reduction of Age models with memory m (ARA m ), proposed by Doyen and Gaudoin 7, defined by theintensity: N t = (t ρ X min(m 1;N t 1) j=0 (1 ρ) j T Nt j) There are also some DHS models for which i 6=1.For example, the model proposed by Wang and Pham 13 can be understood as a DHS model with i =1=ff i 1 and A i = 0. The failure process is a quasi-renewal process: the interfailure times are independent but not identically distributed Other non-dhs models A few other CM models are not included in the DHS family. Other quasirenewal processes can be defined by the intensity : N t = h(n t ;t T Nt ) which is not necessarily of the DHS form. An alternative to the Arithmetic Reduction of Age models is the Arithmetic Reduction of Intensity models (ARI m ), proposed in Doyen and Gaudoin 7, defined by theintensity : N t = (t) ρ X min(m 1;N t 1) j=0 (1 ρ) j (T Nt j) The Chan and Shaw 5 model appears to be a ARI 1 model. Finally, the Trend-Renewal process (Lindqvist 10 ) is defined by a failure intensity of the form : N t = μ(t) z ψz t T N t μ(s)ds!

5 Stochastic modelling of the effect of PM and CM on repairable systems reliability 5 3. Mixing corrective and deterministic preventive maintenance In practice, for important systems of high security, failures are very rare, so there are much more PM than CM actions. Then it is necessary to include preventive maintenance into the above modelling. The basic assumptions are that PM times are planned and deterministic, and that their efficiency obeys to one of the models presented in section 2. If CM are supposed to be ABAO, the failure process is simply an NHPP. For example, the model proposed by Canfield 4, assuming periodic PM with translated ARA efficiency, is defined by the failure intensity : X N t = (t fi t b t tc b t c)+ (i( t fi)+fi) (i( t fi)) i=1 Other models mixing CM and deterministic PM actions have been proposed (Pham and Wang 12 ). 4. Mixing CM and random PM In practice, PM times are not necessarily planned: for example they can be determined according to the state of the failure process or to wear-out controls. Then, models mixing CM and random PM have been proposed. Even if PM times are not deterministic, they are generally assumed to be independent of CM times. In fact, it is desirable to obtain a certain kind of dependence between PM and CM times : PM will be optimal if PM actions are carried out just before" failures. In this section, we propose a modelling mixing CM and random PM, where PM and CM times are not necessarily independent and PM and CM effects are not necessarily the same A virtual age model for PM and CM Let ft i gi 1 and ffi i gi 1 be the CM and PM times. Let fn t gt 0 and fm t gt 0 be the cumulative number of CM and PM. Let fx i gi 1 and fχ i gi 1 be the inter-cm and inter-pm times. We propose a generalized virtual age model including PM and CM effects, defined by the failure intensity: N t (A; M) = (A Mt;N t + t max(fi Mt ;T Nt )) In this model, the virtual age at time t is equal to the time elapsed since the last maintenance: t max(fi Mt ;T Nt ), plus the effective age: A Mt;N t.

6 6 L. Doyen and O. Gaudoin The effective age represents the virtual age just after maintenance and is a function of the previous observed PM and CM times. Most of basic assumptions on the effect of PM and CM can be expressed by a particular form of effective age. For example, ABAO PMandCM are obtained when the effective age is equal to the last maintenance time, A Mt;Nt = max(fi Mt ;T Nt ) and so the failure intensity is simply that of a NHPP: N t (M) = (t) With an effective age equal to zero, PM and CM are AGAN but the failure process is not a renewal process: N t (M) = (t max(fi M t ;T Nt )) We can also have different effects for PM and CM. For example, to have ABAO PM (resp. CM) and AGAN CM (resp. PM) the effective age have to be equal to the time elapsed since the last CM (resp. PM), and the failure intensity is then equal to: N t (M) = (t T N t ) resp. N t (M) = (t fi N t ) By analogy with repair models presented in section 2, we can build models mixing CM and PM effect on the failure process. For example, we can choose a proportional age reduction model for PM and CM with the same improvement factor ρ: N t (M) = (t ρ max(fi Mt ;T Nt )) We can also have different improvement factors ρ P for PM and ρ C for CM. The failure intensity is then equal to: X N t (M) = M t N t t ρ P (fi j maxffi j 1 ;T Nfij g) ρ C j=1 X (T i maxffi MTi ;T i 1 g) In this approach, in which only the failure intensity relative to the PM process is given, the PM-CM process is not completely defined: more information on PM times and dependence between PM and CM times is needed. i= Stochastic modelling of the PM-CM process A competing risk approach A convenient way for modelling the PM-CM process is the competing risk approach, developped for example in Bedford and Mesina 1. When the system is restored after maintenance, the time to next maintenance is either a

7 Stochastic modelling of the effect of PM and CM on repairable systems reliability 7 CM time or a PM time. Let fw i gi 1 denote the times between successive maintenances. Let Y i be the time to next PM in case that no CM occur before and Z i be the time to next CM in case that no PM occur before. Then, the observations are : W i = min(y i ;Z i ) and U i =1I fyi»z ig where U i indicates if the i th maintenance is a CM or a PM. In the classical competing risk problem, each maintenance is supposed to be AGAN and the marginal distributions of Y and Z are studied. The main result is that observations do not allow to estimate these distributions without making additional non-testable assumptions. The approach developped in this paper is very different: we are not interested in the distribution of Y and Z, but in the distribution of the observed CM and PM processes. Then, we can make assumptions on maintenance effect much larger than AGAN Notations Let us recall or introduce all the notations needed in the following. For the global maintenance process (PM and CM): fu i gi 1 the indicators of types of maintenance, U i =1I fyi»z ig fw i gi 1 the times between maintenances fc i gi 1 the maintenance times, C i = P i j=1 W j fk t gt 0 the counting maintenance process, K t =maxfk 2 INjC k» tg For the failure or CM process: fn t gt 0 the counting CM process, N t = P K t k=1 U k ft i gi 1 the CM times, T i =minft 2 IR + jn t = ig fx i gi 1 the times between CM, X i = T i 1 T i Finally for the PM process: fm t gt 0 the counting PM process, M t = P K t k=1 (1 U k) ffi i gi 1 the PM times, fi i =minft 2 IR + jm t = ig fχ i gi 1 the times between PM, χ i = fi i 1 fi i Figure 1 presents an example of a PM-CM process with its corresponding notations.

8 8 L. Doyen and O. Gaudoin K t Y Y Y Y Z 1 Z 2 Z3 Z Λ N t ff -ff -ff -ff Λ C 1 C 3 C 2 W 1 W 2 W 3 W 4-2 M t ff -ff -ff 6 T 1 X 1 X 2 X 3 Λ T 2-1. ff fi 1 -ff. χ 1 χ Fig. 1. Example of a PM-CM process defined by itsy and Z processes with the corresponding PM and CM times

9 Stochastic modelling of the effect of PM and CM on repairable systems reliability 9 Remark: The indicator of maintenance type U i assimilates the case where Y i = Z i to a PM time, that is to say, if there is a failure at a preventive maintenance time, then the failure is not detected and only PM is observed Characterization of the PM-CM process In order to characterize the PM-CM process, we define three intensities, relative to the whole maintenance history. Definition 1: The CM intensity: N 1 t (K; U) = lim dt!0 dt P (N t+dt N t =1jK t ;W 1 ;U 1 :::; W Kt ;U Kt ) The PM intensity: M 1 t (K; U) = lim dt!0 dt P (M t+dt M t =1jK t ;W 1 ;U 1 :::; W Kt ;U Kt ) The (global) maintenance intensity: K 1 t (U) = lim dt!0 dt P (K t+dt K t =1jK t ;W 1 ;U 1 :::; W Kt ;U Kt ) The maintenance intensity can be easily deduced from PM and CM intensities thanks to the following property. Lemma 2: K t (U) = N t (K; U)+ M t (K; U) A generalized sub-survival function The basis of the classical competing risk approach is the study of the subsurvival functions defined as: S Λ Y (t) =P (Y >t;y<z)ands Λ Z(t) =P (Z >t;z<y) We generalize this notion to our context. Definition3: For a PM-CM process we define two sub-survival functions: S Λ Z k+1 (t; w 1 ; :::; u k )=P (Z k+1 >t;z k+1 <Y k+1 jw 1 = w 1 ; :::; U k = u k ) = P (W k+1 >t;u k+1 =0jW 1 = w 1 ;:::;U k = u k ) S Λ Y k+1 (t; w 1 ; :::; u k )=P (Y k+1 >t;y k+1» Z k+1 jw 1 = w 1 ; :::; U k = u k ) = P (W k+1 >t;u k+1 =1jW 1 = w 1 ;:::;U k = u k )

10 10 L. Doyen and O. Gaudoin These functions represent the probability that the next maintenance will not take place within the next t unit of times and that it will be a corrective (resp. preventive) maintenance, when the whole past of the maintenance process is known. They completely define the joint distribution of the observations,thatistosay the processes fw i gi 1 and fu i gi 1. These two sub-survival functions can also be expressed with the two PM-CM intensities. Lemma 4: S Λ Z k+1 (t; w 1 ; :::; u k )= Z +1 t N c k+v(k; w 1 ;:::;u k )e R v 0 K c k +s (k;w1;:::;uk)ds dv S Λ Y k+1 (t; w 1 ; :::; u k )= Z +1 t M c k+v(k; w 1 ; :::; u k )e R v 0 K c k +s (k;w1;:::;uk)ds dv where c k = P k j=1 w j Therefore, the two PM-CM intensities completely define the distribution of the observations. For the demonstration of the model properties, we have to define the notion of sub-density function. Definition 5: The sub-densities functions of a PM-CM process are: f Λ k+1;0(t; w 1 ; :::; u k )= d dt SΛ Z k+1 (t; w 1 ;:::;u k ) f Λ k+1;1(t; w 1 ; :::; u k )= d dt SΛ Y k+1 (t; w 1 ; :::; u k ) We easily deduce from the previous lemma that the sub-densities can be expressed as functions of the PM-CM intensities. Corollary 6: f Λ k+1;0 (t; :::)= N c k+t (k; :::)e R t 0 K c k +s (k;:::)ds f Λ k+1;1(t; :::)= M c k+t(k; :::)e R t 0 K c k +s (k;:::)ds

11 Stochastic modelling of the effect of PM and CM on repairable systems reliability Distribution of the time to the next failure PM and CM intensities can be used to determine the distribution of the time to the next failure, when the whole maintenance history at the time of the last failure is given. This distribution is defined by the probability: Lemma 7: S X;k (x; w 1 ;:::;1) = P (X Nck +1 >xjw 1 = w 1 ; :::; U k =1) S X;k (x; w 1 ;:::;1) = +1X Z jy j=0 i=1 ::: Z P ji=1 y i<x S Wk+j+1 (x jx i=1 y i ;w 1 ; :::; 1;y k+1 ; 0; :::; y k+j ; 0) f Λ k+i;1(y k+1 ;w 1 ; :::; 1;y k+1 ; 0;:::;y k+i 1 ; 0) dy k+j :::dy k+1 where S Wk+1 (x; w 1 ; :::; u k )=S Λ Y k+1 (x; w 1 ;:::;u k )+S Λ Z k+1 (x; w 1 ; :::; u k ) Likelihood function In a parametric approach, we can estimate the parameters of the PM and CM intensities with the likelihood function. This function has a simple expression depending on PM and CM intensities. Theorem 8: The likelihood function corresponding to the observation of a PM-CM process over [0;t] is: L( ; k; w 1 ;:::;u k )= " ky i=1 ui c i (i 1;w 1 ; :::; u i 1 ) # e k+1z cj X j=1 c j 1 K s (j 1;w 1 ; :::; u j 1 )ds ρ where c 0 =0;c k+1 = t; u N t (:::) = t (:::) ifu =1 M t (:::) ifu = Example 1:PM AGAN and CM AGAN As a particular case of our model, AGAN PM and CM represent the classical competing risk approach. In this approach, used by Bedford and Mesina 1, the model is completely defined by the joint distribution of Y and Z: S(y; z) =P (Y 1 >y;z 1 >z)

12 12 L. Doyen and O. Gaudoin But it is well known that the observations do not allow to estimate this function. In fact the two PM-CM intensities only depend of the value of this joint distribution function around the identity line: Theorem 9: For the classical competing risk model, we have: N t (K; U) = sz (t C Kt ) S(t C Kt ) ; M t (K; U) = sy (t C Kt ) S(t C Kt ) where S(t) =S(t; t); s Y (t) = s Z S(y; z) S(y; z) (t; t) and Then, the sub-densities, the distribution of time to the next failure and the likelihood function can be derived. Corollary 10: For the classical competing risk model: f Λ 0 (z) = s Z (z); S X (x) = " S Ω +1X k=0 f Λ 1 (z) = s Y (z) ( 1) k s Y Ωk # (x) in the caseoftimetruncated data L( ; k; w 1 ;u 1 ; :::; w k ;u k )=S(t c k ) ky j=1 (s Y (w j )) uj (s Z (w j )) 1 uj Example 2: ABAO PM and ABAO CM Any kind of assumptions on maintenance effect can be made. For example, a basic case is when PM and CM are ABAO. Theorem 11: When PM and CM are ABAO, the PM-CM intensities are: N t (K; U) = sz (t) ; M t (K; U) = sy (t) S(t) S(t) Then both observed PM and CM processes are NHPP that only depend on the value of the joint distribution function of Y and Z around the identity line. The following quantities are easily obtained. Corollary 12: f Λ 0 (z; c) = sz (z + c) ; f Λ 1 (z; c) = sy (z + c) S(c) S(c)

13 Stochastic modelling of the effect of PM and CM on repairable systems reliability 13 S X (x; c) = S(x + c) S(c)»Z ck+x exp sy (u) c k S(u) du in the case of time truncated data L( ; k; w 1 ;u 1 ; :::; w k ;u k )=S(t) ky j=1 (s Y (c j )) uj (s Z (c j )) 1 uj S(c j 1 ) Example3:PAR PM and PAR CM Finally, the model is applied to the case where PM and CM effect are of the proportional age reduction type. Theorem 13: When PM and CM are PAR, the PM-CM intensities are: N t (K; U) = sz (t ρc Kt ) S(t ρc Kt ) ; M t (K; U) = sy (t ρc Kt ) S(t ρc Kt ) As particular cases, both previous models corresponds respectively to ρ = 1 and ρ = 0. All the following quantities can be derived. Lemma 14: f Λ 0 (z; c) = sz (z +(1 ρ)c) ; f Λ 1 S((1 ρ)c) (z; c) = sy (z +(1 ρ)c) S((1 ρ)c) S X (x; c) = +1X Z j=1 ::: Z where y 0 = c 1 S(c + x ρc)+ S((1 ρ)t) jy S(c + x ρy j ) c»y 1»:::»y j»c+x i=1 for time truncated data L( ; k; w 1 ;u 1 ;:::;w Q k ;u k )= k S(t ρc k ) j=1 f Λ 1 (y i ρy i 1 ) S((1 ρ)y i ) dy j:::dy 1 (s Y (c j ρ c j 1)) uj (s Z (c j ρ c j 1)) 1 u j S((1 ρ) c j 1)

14 14 L. Doyen and O. Gaudoin 5. Conclusion This study proposes a new framework to model possibly linked random PM and CM. Thanks to this modelling, it is possible to establish links between different approaches such as imperfect repair modelling and competing risk problem, and then develop new results. The likelihood function of the model has been derived. We now have to assess the quality of the corresponding estimators and develop goodnessof-fit tests for the models. The final goal is to apply this modelling to real data and estimate maintenance efficiency on complex repairable systems. References 1. T. Bedford and C. Mesina, The impact of modeling assumptions on maintenance optimization, in 2nd Int. Conf. on Math. Meth. in Rel.,(MMR, Bordeaux, 2000), p J. F. Brown, J. F. Mahoney and B. D. Sivazlian, Hysteresis repair in discounted replacement problems, IIE Trans. 15, 2, p (1983). 3. M. Brown and F. Proschan, Imperfect repair, J. of Appl. Prob. 20, p (1983). 4. R. V. Canfield, Cost optimisation of periodic preventive maintenance, IEEE Trans. on Rel. 35, p (1986). 5. J. K. Chan and L. Shaw, Modeling repairable systems with failure rates that depend on age and maintenance, IEEE Trans. on Rel. 42, p (1993). 6. C. Dorado, M. Hollander and J. Sethuraman, Nonparametric estimation for a general repair model, The Ann. of Stat. 25, p (1997). 7. L. Doyen and O. Gaudoin, Modelling and assessment of maintenance efficiency for repairable systems, 13th Eur. Saf. and Rel. Int. Conf. (ESREL, 2002), Lyon, p M. Kijima, Some results for repairable systems with general repair, J. of Appl. Prob. 26, p (1989). 9. M. Kijima, H. Morimura and Y. Suzuki, Periodical replacement problem without assuming minimal repair. Eur. J. of Oper. Res. 37, p (1988). 10. B. Lindqvist, Statistical modelling and analysis of repairable systems, in Statistical and Probabilistic Models in Reliability, Eds. D. C. Ionescu and N. Limnios, (Birkhaüser, Boston, 1999), p M. A. K. Malik, Reliable preventive maintenance policy, AIIE Trans. 11, p (1979). 12. H. Pham and H. Wang, Imperfect maintenance, Eur. J. of Oper. Res. 94, p (1996). 13. H. Wang and H. Pham, A quasi-renewal process and its application in imperfect maintenance, Int. J. of Syst. Sc. 27, p (1996).