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Generalized reliability models and preventive maintenance policy for systems subject to competing dependent failure processes
Proceedigs Proceedigs of of the the 15th 15th IFAC IFAC Symposium Symposium on on Proceedigs the IFAC on Information Control Problems in Manufacturing Manufacturing Available online at www.sciencedirect.com Proceedigs of of the 15th 15th IFAC Symposium Symposium on Information Control Problems in Information Control Problems in Manufacturing Proceedigs of the 15th IFAC Symposium on May 11-13, 2015. Ottawa, Canada Information Control Problems in Manufacturing May 11-13, 2015. Ottawa, Canada May Ottawa, Canada Information Control Problems in Manufacturing May 11-13, 11-13, 2015. 2015. Ottawa, Canada May 11-13, 2015. Ottawa, Canada
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IFAC-PapersOnLine 48-3 (2015) 2164–2169
Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for systems subject to competing dependent failure processes systems reliability subject to models competing dependent failure processes Generalized and preventive maintenance policy for systems subject to competing dependent failure processes ** *** **** systems subject to competing dependent failure processes L. L. Tlili*, Tlili*, M. M. Radhoui Radhoui** A. Chelbi Chelbi*** N. Rezg Rezg**** **,, A. ***,, N. ****
**, A. Chelbi***, N. Rezg**** L. Tlili*, M. L.Ecole Tlili*,Nationale M. Radhoui Radhoui , A. Chelbi , N. Rezg of Tunis, Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, ** *** **** L. Tlili*, M. Radhoui , A. Chelbi , N. Rezg University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected]) Tunisia (email: [email protected]) * ** University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected]) ** Tunisia (email: [email protected]) University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, ** **University Tunisia (email: [email protected]) University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, Tunisia (email: [email protected]) Tunisia (email: [email protected]) ** *** University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, Tunisia (email: [email protected]) ***University Tunisia (email: [email protected]) of Tunis, Tunis, Ecole Ecole Nationale Nationale Supérieure Supérieure d’Ingénieurs d’Ingénieurs de de Tunis, Tunis, CEREP, CEREP, Tunis, Tunis, of *** ***University Tunisia (email: [email protected]) University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected] Tunisia (email: [email protected] *** **** University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected] ****University Tunisia (email: [email protected] of Lorraine, LGIPM, Metz, France ([email protected]) University of Lorraine, LGIPM, Metz, France ([email protected]) **** ****University of Lorraine, Tunisia (email: [email protected] LGIPM, Metz, France ([email protected]) University of Lorraine, LGIPM, Metz, France ([email protected]) **** University in of Lorraine, Franceanalysis ([email protected]) Abstract: We investigate this systems reliability using Abstract: We investigate in this paper paperLGIPM, systemsMetz, reliability analysis using shock shock models. models. Two Two known known Abstract: We investigate in this paper systems reliability analysis using shock models. Two known Abstract: We investigate in this paper systems reliability analysis using shock models. Two known modeling approaches are considered: the first one is a mixed shock model and the second one is based on modeling approaches are considered: the first one is a mixed shock model and the second one is based on Abstract: We investigate in this paper systems reliability analysis using shock models. Two known modeling approaches are considered: the first one is aa mixed shock model and the second one is based on modeling approaches are considered: the first one is mixed shock model and the second one is based on mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, modeling approaches considered: thewhich first one isof a mixed shock model and the second one isprocess. based on mixed degradation and shocks. We generalize each these two models considering aa vulnerability level, mixed degradation andare shocks. We generalize of these models considering vulnerability level, L(t), is magnitude under shock has effect on system A L(t), which which is aaa shock shock magnitude under which aaaeach shock has no notwo effect on the the system degradation degradation process. A mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, L(t), which is shock magnitude under which shock has no effect on the system degradation process. A L(t), which is a shock magnitude under which a shock has no effect on the system degradation process. A preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. A L(t), which maintenance is model a shock magnitude whichdetermine a shock has no effectto the system degradation process. preventive maintenance policy is isunder proposed for systems subject toonmixed mixed degradation and the shocks. A preventive policy for systems subject degradation and shocks. A mathematical in order simultaneously the age optimal mathematical model is is developed developed inproposed order to to determine determine simultaneously the optimal optimal age T* T* and and the optimal preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. A mathematical model is developed in order to simultaneously the optimal age T* and the optimal mathematical model is developed in order to determine simultaneously the optimal age T* and the optimal number of N* which maintenance action be undertaken (whichever comes number of shocks shocks N*isat atdeveloped which aaa preventive preventive maintenance action should shouldthe beoptimal undertaken (whichever comes mathematical model in ordermaintenance to determine simultaneously age T* and the optimal number of shocks N* at which preventive maintenance action should be undertaken (whichever comes number of shocks N* at which a preventive maintenance action should be undertaken (whichever comes first), minimizing the average long-run cost per time unit. Obtained numerical results are first), minimizing minimizing the average long-run maintenance cost per per time unit. Obtained numerical results are number of shocks the N* at which long-run a preventive maintenance should undertaken (whichever comes first), average maintenance cost unit. numerical results are first), minimizing the average long-run maintenance costaction per time time unit.beObtained Obtained numerical results are discussed. discussed. first), minimizing the average long-run maintenance cost per time unit. Obtained numerical results are discussed. discussed. Keywords: Competing failures, degradation, preventive maintenance, shocks © 2015, IFAC (International Federation of Automatic Control) Hosting byreliability, Elsevier Ltd. All rights reserved. Keywords: Competing failures, degradation, preventive maintenance, reliability, shocks discussed. Keywords: Keywords: Competing Competing failures, failures, degradation, degradation, preventive preventive maintenance, maintenance, reliability, reliability, shocks shocks Keywords: Competing failures, degradation, preventive Systems maintenance, shocks have individual variations Systems havereliability, individual variations in in their their ability ability to to Systems have individual variations in their ability to Systems have individual variations in their ability to 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by Systems have individual variations their ability to 1. withstand shocks. Their to istypes: influenced by 1. INTRODUCTION INTRODUCTION withstand shocks. Their resistance to shocks shocks influenced by many factors which canresistance be classified classified inintwo internal many factors which can be in twois types: internal 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by many factors which can be classified in two types: internal manyexternal. factors which can be classified in two internal and The factors due the The and external. The internal internal factors are are duetypes: the intrinsic intrinsic The failure failure of of various various systems systems and and equipment equipment are are generally generally manyexternal. factors which can be classified in two types: internal and external. The internal factors are due the intrinsic The failure of various systems and equipment are generally and The internal factors are due the intrinsic The failure of various systems and equipment are generally characteristics of the system (e.g. the quality of material, the due to competing failure modes. The first one is what is characteristics of the system (e.g. the quality of material, the due to competing competing failure modes. The first one one isgenerally what is is and external. The internal factors are due the intrinsic characteristics of the system (e.g. the quality of material, the The failure of various systems and equipment are due to failure modes. The first is what characteristics of the system (e.g. the quality of material, the due to competing failure modes. The first one is what is system structure and design, etc.). However, the external called catastrophic failure following some sudden external system structure and design, etc.). However, the external called catastrophic failure following some sudden external characteristics of the the However, quality of material, the system structure and design, etc.). the due to catastrophic competing failure modes. Thesome first one is external what is called failure following sudden system structure andsystem design,(e.g. the external external called catastrophic failureto some factors are generally related to the operation shocks due overheating or random voltage factors are generally related toetc.). the However, operation environment environment shocks due for for example example tofollowing overheating or sudden random external voltage system structure and design, etc.). However, the external factors are generally related to the operation environment called catastrophic failure following some sudden external shocks due for example to overheating or random voltage factors are generally related to the operation environment shocks due for example to overheating or random voltage (e.g. temperature, humidity, pressure, etc.). spikes or sudden and usage or (e.g. temperature, humidity, pressure, etc.). spikes or sudden and unexpected unexpected usage loads, loads, or accidental accidental factors are generally related tothethecumulative operation environment (e.g. temperature, humidity, pressure, etc.). shocks or due for hard example to overheating or random voltage spikes sudden and unexpected usage loads, or accidental (e.g. temperature, humidity, pressure, etc.). spikes or sudden and surfaces. unexpected usage loads, or is accidental In the literature, in the case of shock dropping onto The second one due to In the literature, in the case of the cumulative shock model, model, dropping onto hard surfaces. The second one is due to (e.g. temperature, humidity, pressure, etc.). In the literature, in the case of the cumulative shock model, spikes or sudden and unexpected usage loads, or accidental dropping onto hard surfaces. The second one is due to In the literature, in the assumed case of the cumulative shock model, dropping onto hard surfaces. The second onemodels is due are to the system is usually to be vulnerable to shocks cumulative physical deterioration. Shock the system is usually assumed to be vulnerable to shocks cumulative physical deterioration. Shock models are In the literature, infalls the assumed case of the cumulative shock model, the system is usually to be vulnerable to shocks dropping onto hard surfaces. The second onemodels is due are to cumulative physical deterioration. Shock the system is usually assumed to be vulnerable to shocks cumulative physical deterioration. Shock models are whose magnitude between 0 and a critical failure level. appropriate for modeling the reliability of systems subject to whose magnitude falls between 0 and a critical failure level. appropriate for modeling the reliability of systems subject to the system is usually assumed toand be to fact, shocks whose magnitude falls between 0 aavulnerable critical failure level. cumulative physical deterioration. Shock models are appropriate for modeling the reliability of systems subject to whose magnitude falls between 0 and critical failure level. appropriate for modeling the reliability of systems subject to Obviously, this assumption is not usually true. In in these kind of competing failure modes. Obviously, this assumption is not usually true. In fact, in these kind of competing failure modes. whose situations, magnitude falls between 0 and amagnitude critical level. Obviously, this assumption assumption is not not usually true.failure In fact, in appropriate modelingfailure the reliability these kind competing modes. Obviously, this is usually true. In fact, in these kind of offor competing failure modes. of systems subject to many shocks with small may have many situations, shocks with small magnitude may have Obviously, this assumption is not usually true. In fact, in many situations, shocks with small magnitude may have these kind of competing failure modes. studied by many many situations, with small magnitude may have negligible effect the process. Hence, it Shock negligible effect on onshocks the degradation degradation process. Hence, it would would Shock models models have have been been extensively extensively studied studied by by many many many situations, shocks with small magnitude may have negligible effect on the degradation process. Hence, it would Shock models have been extensively negligible effect on the degradation process. Hence, it would Shock models have been extensively studied by many be relevant to aa certain of magnitude, researchers (Chelbi et in to be relevanteffect to consider consider certain level level of shock shock magnitude, researchers (Chelbi et al., al., 2000)) 2000)) in order order to determine determine negligible on system the degradation process. Hence, it would be relevant to aa certain level of shock magnitude, Shock models have et extensively studied by These many researchers (Chelbi al., 2000)) in to be relevant to consider consider certain level oflevel, shock magnitude, researchers (Chelbi etbeen al., of 2000)) in order order to determine determine which we will call vulnerability below which mathematical expressions of systems reliability. which we will call system vulnerability level, below which mathematical expressions systems reliability. These be relevant to consider a certain level of shock magnitude, which we will call system vulnerability level, below which researchers (Chelbi et al., 2000)) in order to determine mathematical expressions of systems reliability. These which we will call system vulnerability level, below which mathematical expressions of systems reliability. These there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): whichis weno will callon system vulnerability level, below which there effect the system degradation process. This mathematical expressions ofcategories systems reliability. These models are classified into four (Peng et al., 2011): there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude there is no effect oncan the system degradation process. This vulnerability level be estimated from shock and models are classified into four categories (Pengthe et al., 2011): (i) extreme shock model: failure occurs when magnitude vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude deterioration data with classical statistical methods. of any shock exceeds some critical level (called resistance to deterioration data with classical statistical methods. of any shock exceeds some critical level (called resistance to vulnerabilitythelevel can beclassical estimated from shock and deterioration data with statistical methods. (i) any extreme shock model: occurs magnitude of shock exceeds some critical level (called resistance to deterioration with classical statistical methods. of any shock somefailure critical level when (calledthe resistance to Moreover, system vulnerability and its to shock) (Gut et (ii) shock model: Moreover, the data system vulnerability and its resistance resistance to shock) (Gut exceeds et al., al., 1999); 1999); (ii) cumulative cumulative shock model: deterioration data with classical statistical methods. Moreover, the system vulnerability and its resistance to of any shock exceeds some critical level (called resistance to shock) (Gut et al., 1999); (ii) cumulative shock model: Moreover, the system vulnerability and its resistance to shock) (Gut et al., 1999); (ii) cumulative shock model: shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks Moreover, system its resistance to shocks may the decrease as it itvulnerability gets older older and andand degrades. shock) occurs (Gut etwhen al., 1999); (ii)(Gut, cumulative shock model: failure the damage from shocks shocks may decrease as gets degrades. failure occurs when the cumulative cumulative damage(iii) from exceeds some threshold run shock exceeds some given given threshold (Gut, 1990); 1990); (iii) runshocks shock shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks exceeds some given threshold (Gut, 1990); (iii) run shock exceeds some given threshold (Gut, 1990); (iii) run shock In model: occurs when is of In this this paper, paper, we we propose propose two two mixed mixed shock shock models models to to model model model: failure failure occursthreshold when there there is aaa run run(iii) of kkk shocks shocks In we two mixed models exceeds some given (Gut, is shock model: failure occurs when run of shocks model: failure occurs when there there is1990); a al., run2001); of run k and shocks In this this paper, paper, we propose propose two account mixed shock shock models to to model model systems reliability taking into the vulnerability level exceeding aa critical magnitude (Mallor et (iv) systems reliability taking into account the vulnerability level exceeding critical magnitude (Mallor et al., 2001); and (iv) In this paper, we propose twonoaccount mixed shock models to model systems reliability taking into the vulnerability level model: failure occurs when there isthe a al., run2001); of kbetween shocks exceeding aa critical magnitude (Mallor et and (iv) exceeding critical magnitude (Mallor et al., 2001); and (iv) systems reliability taking into account the vulnerability level below which shocks have effects on the degradation δ-shock model: failure occurs when time lag below which shocks have no effects on the degradation δ-shock model: failure occurs when the time lag between systems reliability taking intono account theon vulnerability level below which shocks have effects the degradation exceedingmodel: ashocks critical magnitude (Mallor et al., 2001); andet(iv) δ-shock failure occurs when the time lag between δ-shock model: failure occurs when the time lag between below which shocks have no effects on the degradation process. The first model combines the extreme shock and successive is shorter than a threshold δ (Wang al. process. The first first modelhave combines the extreme extreme shock and the the successive shocksfailure is shorter shorter thanwhen threshold (Wang et al. al. below which shocks no cumulative effects on degradation theshock degradation process. The model combines the and the δ-shock model: occurs the timeδδδ (Wang lag between successive shocks is than aaa threshold et successive shocks is shorter than threshold (Wang et al. process. The first model combines the extreme shock and due the cumulative shock models with 2001). cumulative shock models with cumulative degradation due 2001). process. Theshock first model combines theanextreme shock andfirst the cumulative models with cumulative degradation due successive shocks is shorter thanofa threshold δ (Wang et al. 2001). 2001). cumulative shock models with cumulative degradation due only to shocks. The second model is extension of the Furthermore, different mixtures these models have been only to shocks. shocks. Themodels secondwith model is an an extension extension of the the first first Furthermore, different mixtures mixtures of of these these models models have have been been cumulative shock cumulative due only The model is of 2001). Furthermore, different Furthermore, different mixtures of they theseare models havecalled been only to to shocks. The second second model is due an extension of the one with cumulative degradation to degradation shocks andfirst to investigated by several researchers; generally one with cumulative degradation due to shocks and to investigated by several researchers; they are generally called only to shocks. The second model is an extension of the first one with cumulative degradation due to shocks and to Furthermore, different mixtures of these models have been investigated by several researchers; they are generally called investigated by several researchers; they are generally called one with cumulative degradation due to shocks and to internal factors. factors. mixed shock models. Shanthikumar et al. (1983) proposed aa internal mixed shock models. Shanthikumar et al. (1983) proposed one with cumulative degradation due to shocks and to internal factors. investigated by several researchers; they are generally called mixed shock models. Shanthikumar et al. (1983) proposed a mixed shock models. Shanthikumar et al. (1983) proposed a internal factors. general model which the shock model general modelmodels. which combines combines the extreme extreme shock proposed model and and internal mixed shock Shanthikumar etbased al. (1983) general model which combines the shock model and general model which combines the extreme extreme shock model anda We also alsofactors. propose aa preventive preventive maintenance maintenance (PM) (PM) policy policy for for δ-shock model. This combination is on the fact that We propose δ-shock model. This combination is based on the fact that We also propose aa preventive maintenance (PM) policy for general model which combines the extreme shock model and δ-shock model. This combination is based on the fact that δ-shock model. This combination is based on the fact that We also propose preventive maintenance (PM) policy for systems whose reliability is modeled according to the second the magnitude of the k-th shock and the time interval systems whose reliability is modeled according to the second the magnitude of thecombination k-th shock shockis and and theontime time interval We alsoshock propose a preventive maintenance (PM) policy for systems whose reliability is modeled according to the second δ-shock model. Thisthe basedthe the fact that the magnitude of k-th interval the magnitude of the k-th shock and the time interval systems whose reliability is modeled according to the second mixed model involving degradation due to internal between consecutive shocks are correlated with each other. mixed shock model involving degradation due internal between consecutive shocks are correlated with each each other. systems whose reliability is modeled according to to the second mixed shock model involving degradation due to internal the magnitude of the k-thare shock and the interval between consecutive shocks correlated with between consecutive shocks are correlated with time each other. other. mixed shock model involving degradation due to internal and external factors. System is replaced replaced following failure or and external factors. System is following failure or mixed shock model involving degradation due to internal and external factors. System is replaced following failure or between consecutive shocks are correlated with each other. and external factors. System is replaced following failure or 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 2238 and external factors. System is replaced following failure or ** University **University
Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 2238Control. Copyright © 2238 Copyright © 2015 2015 IFAC IFAC 2238 10.1016/j.ifacol.2015.06.409 Copyright © 2015 IFAC 2238
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preventively at an age T or after having been subject to N shocks. A mathematical model is developed to determine the optimal age T* and the number of shocks N*, which minimize the average long-run maintenance cost per time unit.
(i.e. the time from any time t to the next shock is independent of time t). Shocks occur with a stationary rate 𝜆𝜆. Let N(t) denote the number of random shocks until time t. Then the probability Hi(t) that shocks occur exactly i times in time interval (0, t] is given by: H i t P N t i
In the literature, there are many maintenance policies based on shock models. Wang et al. (2001) consider a system whose failure is induced by the δ-shock model. They determine the optimal failure number N* such that the longrun average cost per time unit is minimized. This work has been extended by Wang et al. (2005) considering a mixed shock model (i.e. extreme shock model and δ-shock model). They consider a repairable system subject to two types of failure. The first one is the result of the inter-arrival time between consecutive shocks being shorter than a given threshold value. The second one is based on the magnitude of a single shock that is higher than a given threshold level value. Nakagawa (1976) considers the cumulative shock model. He proposes a replacement policy according to which an item is replaced at failure or when a certain level of cumulative damage is reached, whichever occurs first. A mathematical framework is developed to obtain the optimal level of damage which minimizes the long-run cost per unit of time. Nakagawa et al. (1989) propose a periodic replacement policy with minimal repair at failure considering the cumulative shock model. They obtain the optimal solution for the time T*, number of shocks N*, and damage level Z* at which preventive replacement should be performed. Qian et al. (2005) proposed a maintenance policy considering an extended cumulative shock model with shocks occurring according to a non-homogeneous Poisson process. The decision variables are the failures number N and the age T.
t
i
(1)
e t
i! (B) Shocks follow a non-homogeneous Poisson Process (NHPP) when shocks arrival is not stationary. In such situations, if t stands for the shocks occurrence rate, then the probability Hi(t) that shocks occur exactly i times in time interval (0, t] is given by:
t e i
Hi t
t
i!
(2)
t
Where t u du 0
If shocks occur at times T1, T2, …, according to NHPP, then the probability that more than (i+1) shocks occur during (0,t] is:
P T i 1 t H i u u du t
0
H t
(3)
j
j i 1
In this section, we consider a system subject to shocks which occur at times Tj (j = 1, 2,…). The shock magnitudes are denoted by Wi. They are assumed to be random variables, sindependent, and identically distributed with a common distribution GW where GW(0)=0. In order to express the system reliability function, the mixed shock model that is considered here is based on the one which combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks (Gut, 2001)). Failures occur following an extreme shock whose magnitude is higher that a given critical level U (extreme shock model), or whenever cumulative damage exceeds a given degradation level threshold. The total degradation is assumed to be accumulating through shocks whose magnitude is between zero and U (cumulative shock model). In fact, this assumption is not always true. In many situations, shocks with small magnitude (minor shocks) may have no effect on the degradation process. Hence, we introduce a vulnerability level, L, below which shocks have no effect on the system degradation process; they will be called minor shocks. Both levels L and U are either independent of time t or decreasing with respect to operating time (or number of shocks). Keeping them constant means that the system can recover from the consequences of a previous shock (i.e. the mean inter-arrival time of shocks is much larger than the mean time of recovery). In case extreme shock and cumulative shock provoke an increase in the probability of system failure, L(t) and U(t) are considered as decreasing with time (or number of shocks). This is usually the case for many machines and equipment. In this work, we consider the general case (L(t) and U(t): constant or decreasing functions).
The remainder of this paper is organized as follows. Section 2 is dedicated to the development of a generalized reliability expression using a mixed shock model which combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks. In section 3, we develop the generalized expression of system reliability in the case of a mixed degradation and shock model considering cumulative degradation due to shocks and to internal factors. Section 4 is devoted to the definition and mathematical modeling of a preventive maintenance policy. The latter suggests performing PM actions at shock number N or at age T, whichever occurs first. A numerical example is discussed in Section 5. Finally, some comments and conclusions are drawn in the last section. 2.
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MODEL 1: GENERALIZED MIXED SHOCK MODEL
Generally shocks occurrences over time are modeled according to a Poisson process. We distinguish two kinds of Poisson processes: (A) Shocks follow a homogeneous Poisson process (HPP) when the times between successive shocks are distributed exponentially and have a memoryless property. In other words, shocks are generated randomly and uniformly in time 2239
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Our proposed generalized mixed shock model will combine three major types of shocks - namely extreme shock, cumulative shock and minor shock. System failure occurs due to an extreme shock whose magnitude exceeds some given critical level U(t), whereas cumulative shock causes a cumulative damage to system only in case the shock magnitude is higher than the vulnerability level L(t) and lower than the critical level U(t). Failures due to cumulative shocks occur whenever the cumulative damage exceeds a certain threshold level D. Minor shocks (whose magnitude is lower than the vulnerability level L(t)) do not cause any damage to the system. The proposed generalized mixed shock model is summarized as follows considering a given shock of magnitude Wi occurring at time Ti:
Yielding:
R1 D , t H 0 t H i t G si D
Stieltjes convolution of any distribution Gs(x) with itself; Gs(0)(x)=1 for x≥0. 3. MODEL 2: GENERALIZED MIXED DEGRADATION AND SHOCK MODEL In this section, we extend the generalized mixed shock model presented in section 2 considering that cumulative degradation is due not only to shocks but also to internal factors. Thus, the overall damage of the system is cumulative. It is caused by (i) internal degradation through a random process function, X d t which represents the
degradation process causing additional damage X s Ti
deterioration of the system corresponding to working time and (ii) external random shocks inducing cumulative damage expressed by X s Ti , i 0,1, 2,... .
Wi L Ti the shock has no effect on the system
degradation process (minor shock). Events E.1, E.2 and E.3 are mutually exclusive.
Failure occurs whenever the cumulative degradation process exceeds a given threshold level D or shock magnitude exceeds a critical level U(t), whichever occurs first. The cumulative degradation is expressed as: N t
X t X d t X s T j I T , L ,U
i 0,1, 2,... (4)
j 0
R 2 D ,t P X t D
N t P X d t X s T j I T , L ,U D | N t i j i 0 j 0
L t W i U t U t
P N t i
dGWi z
P X d t D ; I T i , L ,U 0; i 0
GWi U t GWi L t
W 1 L t ,W 2 L t ,...,W i L t | N t i P N t i
(5)
N t P X d t X s T j D ; i 1 j 0
The cumulative damage due to random shocks by time t is given as: N t
X t X s Ti I Ti , L,U
(9)
j
Therefore, the system reliability at time t for a given threshold level D of cumulative damage is given by:
Suppose i shocks occur before time t, the probability that the intensities of each of these shocks are between L(t) and U(t) is given by: PU , L t P L t W 1 U t , L t W 2 U t ,...,
L t
(8)
of X s t and Gs(j)(x) (j = 1, 2, · · · ) denotes the j-fold
E.2 L Ti Wi U Ti the shock occurrence accelerates the
dGWi z
where Gs ( x) is the Cumulative distribution function (Cdf)
system
We consider the following binary variable: 1 if L T i W i U T i I Ti , L ,U 0 if W i L T i Where T0 0 and T0 , L ,U =0
L t
i 1
E.1 U Ti Wi the shock causes immediate failure of the
E.3
U t
L t W 1 U t , L t W 2 U t ,..., L t W i U t | N t i P N t i
(6)
i 0
Hence, the survival distribution of the system can be expressed as follows: R1 D , t P X t D
(10)
N t
which yields the following expression: P X s T j I T , L ,U D | N t i j i 0 j 0 R 2 D , t H 0 t Fd D , t H i t Fd D ,t GWi L t ×P N t =i i 1
D U t N t H i t Fd D u , t dG si u dGWi z P X s T j D ; I T , L ,U 1| N t i L 0 t j i 1 i 0 j 0 (11) ×P N t =i Where Fd ( x, t ) is the Cdf of X d t (7)
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It is interesting to note that the two proposed reliability models are a generalization of several past studies. For instance in case L(t)=0 and U(t)=U (constant and finite): models 1 and 2 are reduced to two types of dependent failure processes i.e. cumulative shock and extreme shock model. Hence, we obtain the following results as obtained by (Pen et al. (2001):
R1 D , t H 0 t H i t G s
i
D GWi U
Our objective is to determine simultaneously the optimal age T*, as well as the number of shocks N* to perform PM such as the average long-run maintenance cost rate is minimum. Using classical renewal arguments, the total average maintenance cost per time unit can be expressed over a renewal cycle S. In fact, as previously stated, the system is considered to be as good as new after all maintenance actions.
(12)
i 1
Hence, the expression of the long-run average maintenance cost per time unit is given by:
R 2 D , t H 0 t Fd D , t
H i t Fd D u , t dG si u GWi U D
EC T , N
0
i 1
(13) 4.
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E CTotal T , N
(14)
E TS T , N
Where E CTotal represents the expected total maintenance
MAINTENANCE POLICY
cost incurred within a renewal cycle, and E TS is the mean operating time (average duration of a renewal cycle). The following analysis will lead to the expression of the average long-run maintenance cost per time unit.
In this section, we propose a preventive maintenance policy for a system subject to the generalized mixed degradation and shock model (model 2 presented in section 3) considering three types of shocks. Minor Shocks with a
relatively low magnitude ( Wi L Ti ) have no effect on
4.1 Expected total cost within a cycle
the system, shocks with a high level of magnitude (
The expected total maintenance cost during a cycle E CTotal can be expressed as follows:
U Ti Wi ) (extreme shocks) cause immediate failure, and
shocks
with
an
intermediate
magnitude
level
(
E CTotal (T , N ) CT PT (N ,T ) C N PN (N ,T ) C D PD (N ,T ) CU PU (N ,T )
L Ti Wi U Ti ) accelerate the degradation process by
some amount X s Ti . The cumulative damage is due to
internal degradation and external random shocks. The considered system experiences two competing dependent failure modes: type-I failure and type-II failure. Type-I failure occurs when cumulative degradation exceeds a given threshold level D. Type-II failure occurs due an extreme shock. The system is replaced preventively before failure at an age T (type-I PM action), or after having been subject to a number of shocks N (type-II PM action), whichever occurs first.
(15)
The four terms of the total maintenance cost correspond respectively to type-I PM, type-II PM, type-I CM and type-II CM costs during a renewal cycle. Analytical expressions of each of these different components are developed below. -
The probability PT of having a type-I PM (at age T) is
expressed as follows: N T PT T , N P X d T X s T i T i , L ,U D ; i 0
W 1 U ,W 2 U ,...,W i U ; N T N
Hence, the proposed maintenance policy preconizes the following maintenance actions:
N 1
H i T
- Type-I and type-II PM actions: carried out at age T and at
i 1
shock number N, respectively;
D
0
Fd D u ,T dG si u dGWi z U
L
N 1
H i T Fd D ,T GWi L
- Type-I
corrective maintenance CM action: performed whenever the total damage exceeds a threshold level D.
i 0
(16)
- Type-II
CM action: carried out in case one shock’s magnitude exceeds the critical level U.
-
The probability PN of having a type-II PM (after N
successive shocks) is given by: N PN T , N P X d T X s T i T i , L ,U D ; i 0
Working assumptions: - Random shocks occur according to a Poisson process. - Failures are self-announced and are immediately fixed. - All maintenance actions renew or bring back the system to a state as good as new. - Durations of maintenance actions are negligible. - The costs of Type-I PM (CT), Type-II PM (CN), Type-I CM (CD) and Type-II CM (CU) are constant and known. - The critical level, the vulnerability level and the degradation threshold level are constant.
W 1 U ,W 2 U ,...,W N U ; T N T T
D
U
H N 1 t t dt Fd D u ,T dG s N u dGWN z 0
0
T
L
H N 1 t t dt Fd D ,T G 0
N W
L (17)
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5.
The probability PD of having a type-I CM (an exceeding of threshold level D of cumulative degradation) is given by. j PD T , N P X d T X s T i T i , L ,U D ; i 0 -
NUMERICAL EXAMPLE
In order to discuss the developed reliability functions for both models 1 and 2 and illustrate the application of the maintenance strategy, we consider for instance MicroElectro-Mechanical Systems –MEMS. These systems are j 1 small integrated devices that combine micro-electrical and X d T X s T i T i , L ,U D ; micro-mechanical components and permit the inclusion of i 0 mechanical elements such as gears, diaphragms, and springs W 1 U ,W 2 U ,...,W j 1 U ; j 1 N ;T j 1 T to integrated circuits. We consider a system that contains one microengine. Such a system was studied by Tanner et al. 1 Fd D ,T (2003). The microengine consists of gears connected N 1 T GWi L GW U mechanically to orthogonal linear comb drives and springs. 0 H i t t dt Fd D ,T G i 1 L Rotation of gears is ensured by linear displacement of the i 0 W comb drives. The microengine is assumed to be subject to D F D u ,T dG i u two competing failure processes; wear degradation due to s 0 d aging and debris from shock loads, and spring fracture due to U i a large shock load (Jiang et al. (2012)). N 1 T L dGW z GW U In the following numerical example, time t will be expressed H i t t dt D 0 i 0 Fd D u ,T dG s i 1 u in terms of number of revolutions. Suppose that wear 0 degradation due to aging increases linearly, Xd(t)=x0+βt, U dG i 1 z where β is a normally distributed random variable, β∼N(𝜇𝜇β, W L 𝜎𝜎β2), and wear degradation due to debris from shock loads (18) follows normal distribution Xs∼N(𝜇𝜇s, 𝜎𝜎s2) (Peng et al., 2011). - The probability PU of having type-II CM (extreme We assume also that shocks magnitudes are distributed shock) is given by: exponentially GW t 1 exp t , where θ = 1.2GPa, and j shocks occur according to a Poisson process with stationary PU T , N P X d T X s T i T i , L ,U D , rate λ. i 0 The following input parameters are given below in Table1. W 1 U ,W 2 U ,...,W j U ,W j 1 U ; They are similar to those used by Peng et al. (2011). The obtained results are presented in the next subsections. j 1 N ; T j 1 T N 1 T
H i t t dt Fd D u ,T dG s i u i 1
D
0
Table 1. Input data
0
U
dG L
i W
z U dGW y
H i t t dt Fd D ,T G
i W
0
1.5
L U dGW y
E T S R t , N 1dt T
0
i 1
T
0
N 1 T
H i t Fd D u , t dG s D
0
0
0
𝜎𝜎β
𝜇𝜇s
(𝜇𝜇m3) (𝜇𝜇m3)
8.4823 6.0016 10-9 10-10
10-4
𝜎𝜎s
λ
2 10-5
2.5
(𝜇𝜇m3) (10-5)
U
u dt L dGWi z (20)
R1(D,t)
L=1.5 = U
H i t Fd D , t dtG i 0
i
12.5 10-4
𝜇𝜇β
(𝜇𝜇m3)
Using the input data given above, we computed the reliability functions corresponding to both models 1 and 2 for different values of the system vulnerability level L. These survival functions are presented in figures 3 and 4.
The mean operating time between successive maintenance actions can be expressed as follows: N 1
x0
5.1 Reliability functions behavior relative to the system vulnerability level
4.2 Average duration of a renewal cycle:
D
U
(GPa) (𝜇𝜇m3) (𝜇𝜇m3)
N 1 T i 0
(19)
i W
L
L=1.0 L=0.5
Hence, the average long-run cost rate function CTotal (T , N ) can be obtained by combining equations (16) to (20). The resulting expression is a function of the decision variables which are the age T and the number of shocks N . A numerical procedure is designed to obtain the optimal preventive maintenance policy. The procedure searches for different values of T∊ (0, Tmax] and N∊[1, Nmax] to determine the optimal age T* and the number of shocks N* which minimize the average long-run maintenance cost rate.
L=0.0
t Fig. 3: Reliability behavior in case of model 1 2242
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probability of occurrence of failure, the system is maintained preventively at a certain age T* or after a certain number of shocks N* (whichever occurs first). A mathematical expression of the expected total maintenance cost rate has been developed, and a numerical procedure has been used to determine simultaneously the optimal values of the decision variables T* and N*. The study of run shock model and δshock models when the system vulnerability level is taken into account represent one of the major possible extensions of this work. It is currently under consideration by the authors.
R2(D,t) L=1.5 = U L=1.0 L=0.5 L=0.0
REFERENCES
t
Chelbi, A. and Ait-Kadi, D. (2000) Generalized Inspection Strategy for Randomly Failing Systems Subjected to Random Shocks, International Journal of Production Economics, Vol. 64, pp-379384. Jiang L., Feng Q. and Coit D.W. (2012) Reliability and Maintenance Modeling for Dependent Competing Failure Processes With Shifting Failure Thresholds, IEEE Transactions On Reliability, vol. 61, no. 4. Gut, A. (1990) Cumulative shock models, Advances in Applied Probability 22:504–507. Gut, A. (2001) Mixed shock models, Bernoulli 7(3):541– 555. Gut, A. and Husler, J. (1999) Extreme shock models, Extremes 2, 293-305. Li, ZH. and Kong, XB. (2007) Life behavior of d-shock model, Statistics & Probability Letters 77:577–587. Mallor, F. and Omey, E. (2001) Shocks, runs and random sums, Journal of Applied Probability 38, 438-448. Nakagawa, T. (1976) On a replacement problem of a cumulative damage model. Operational Research Quarterly, 895-900. Nakagawa, T. and M. Kijima (1989). Replacement policies for a cumulative damage model with minimal repair at failure. IEEE Trans. Reliability, 13, 581-584. Peng, H., Feng, Q. M., and Coit D. W. (2011) Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes, IIE Transactions, vol. 43, pp. 12-22. Qian, CH, Ito K and Nakagawa T (2005). Optimal preventive maintenance policies for a shock model with given damage level. Journal of Quality in Maintenance Engineering 11:216–227. Shanthikumar, JG. and Sumita U. (1983) General shock models associated with correlated renewal sequences, Journal of Applied Probability 20:600–614. Tanner, D.M. and Dugger, M.T. (2003) Wear mechanisms in a reliability methodology, Proceedings of the Society of Photo-optical Instrumentation Engineers, 4980, 22–40. Wang, G.J. and Zhang Y.L. (2001) δ-shock model and its optimal replacement policy, Journal of Southeast University, 31, pp. 121–124. Wang, G. J. and Zhan, Y. L. (2005) A shock model with two-type failures and optimal replacement policy, International Journal of Systems Science, 36:4, 209214.
Fig. 4: Reliability behavior in case of model 2 5.2 Optimal PM policy In this subsection, we assume that the vulnerability level is fixed and constant, L=1.0 GPa. The costs of type-I PM, typeII PM, type-I CM and type-II CM have been arbitrarily chosen as: 20$, 20$, 190$ and 250$, respectively. We applied the numerical procedure varying T and N as follows T∊(0,106] and N∊[1, 10]. The obtained optimal solution is presented in Table 2. Table 2. The obtained optimal solution of preventive maintenance policy T* E[TS]* EC* N* PT PN PD PU (revolutions) (revolutions) ($/cycle) 9 104
4
0.61 0.09 0.005 0.30
7.3104
2169
90
Hence, the maintenance crew has to replace the system preventively before failure after T*=9x104 revolutions (typeI PM action), or after having been subject to N*=4 shocks (type-II PM action), whichever occurs first. The corresponding total average maintenance cost is equal to 90$/cycle with an average cycle duration of 7.3x104 revolutions. 6. CONCLUSIONS In this paper, we proposed a generalization of two mixed shock models to express systems reliability taking into account, what we called the system vulnerability level below which shocks have no effects on the degradation process. The first model combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks. The second model is an extension of the first one with cumulative degradation due to shocks and to internal factors. The developed expressions represent a generalization of some previously obtained results in the literature. We also proposed a preventive maintenance policy considering for systems subject to the generalized mixed degradation and shock model. The system is subject to two competing dependent failure modes due to either extreme shock, or the accumulation of damage. To lower the 2243
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Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for Generalized reliability models and preventive maintenance policy for systems subject to competing dependent failure processes systems reliability subject to models competing dependent failure processes Generalized and preventive maintenance policy for systems subject to competing dependent failure processes ** *** **** systems subject to competing dependent failure processes L. L. Tlili*, Tlili*, M. M. Radhoui Radhoui** A. Chelbi Chelbi*** N. Rezg Rezg**** **,, A. ***,, N. ****
**, A. Chelbi***, N. Rezg**** L. Tlili*, M. L.Ecole Tlili*,Nationale M. Radhoui Radhoui , A. Chelbi , N. Rezg of Tunis, Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, ** *** **** L. Tlili*, M. Radhoui , A. Chelbi , N. Rezg University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected]) Tunisia (email: [email protected]) * ** University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected]) ** Tunisia (email: [email protected]) University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, ** **University Tunisia (email: [email protected]) University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, Tunisia (email: [email protected]) Tunisia (email: [email protected]) ** *** University of Carthage, Ecole Supérieure de Technologie et d’Informatique, CEREP, Tunis, Tunisia (email: [email protected]) ***University Tunisia (email: [email protected]) of Tunis, Tunis, Ecole Ecole Nationale Nationale Supérieure Supérieure d’Ingénieurs d’Ingénieurs de de Tunis, Tunis, CEREP, CEREP, Tunis, Tunis, of *** ***University Tunisia (email: [email protected]) University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected] Tunisia (email: [email protected] *** **** University of Tunis, Ecole Nationale Supérieure d’Ingénieurs de Tunis, CEREP, Tunis, Tunisia (email: [email protected] ****University Tunisia (email: [email protected] of Lorraine, LGIPM, Metz, France ([email protected]) University of Lorraine, LGIPM, Metz, France ([email protected]) **** ****University of Lorraine, Tunisia (email: [email protected] LGIPM, Metz, France ([email protected]) University of Lorraine, LGIPM, Metz, France ([email protected]) **** University in of Lorraine, Franceanalysis ([email protected]) Abstract: We investigate this systems reliability using Abstract: We investigate in this paper paperLGIPM, systemsMetz, reliability analysis using shock shock models. models. Two Two known known Abstract: We investigate in this paper systems reliability analysis using shock models. Two known Abstract: We investigate in this paper systems reliability analysis using shock models. Two known modeling approaches are considered: the first one is a mixed shock model and the second one is based on modeling approaches are considered: the first one is a mixed shock model and the second one is based on Abstract: We investigate in this paper systems reliability analysis using shock models. Two known modeling approaches are considered: the first one is aa mixed shock model and the second one is based on modeling approaches are considered: the first one is mixed shock model and the second one is based on mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, modeling approaches considered: thewhich first one isof a mixed shock model and the second one isprocess. based on mixed degradation and shocks. We generalize each these two models considering aa vulnerability level, mixed degradation andare shocks. We generalize of these models considering vulnerability level, L(t), is magnitude under shock has effect on system A L(t), which which is aaa shock shock magnitude under which aaaeach shock has no notwo effect on the the system degradation degradation process. A mixed degradation and shocks. We generalize each of these two models considering a vulnerability level, L(t), which is shock magnitude under which shock has no effect on the system degradation process. A L(t), which is a shock magnitude under which a shock has no effect on the system degradation process. A preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. A L(t), which maintenance is model a shock magnitude whichdetermine a shock has no effectto the system degradation process. preventive maintenance policy is isunder proposed for systems subject toonmixed mixed degradation and the shocks. A preventive policy for systems subject degradation and shocks. A mathematical in order simultaneously the age optimal mathematical model is is developed developed inproposed order to to determine determine simultaneously the optimal optimal age T* T* and and the optimal preventive maintenance policy is proposed for systems subject to mixed degradation and shocks. A mathematical model is developed in order to simultaneously the optimal age T* and the optimal mathematical model is developed in order to determine simultaneously the optimal age T* and the optimal number of N* which maintenance action be undertaken (whichever comes number of shocks shocks N*isat atdeveloped which aaa preventive preventive maintenance action should shouldthe beoptimal undertaken (whichever comes mathematical model in ordermaintenance to determine simultaneously age T* and the optimal number of shocks N* at which preventive maintenance action should be undertaken (whichever comes number of shocks N* at which a preventive maintenance action should be undertaken (whichever comes first), minimizing the average long-run cost per time unit. Obtained numerical results are first), minimizing minimizing the average long-run maintenance cost per per time unit. Obtained numerical results are number of shocks the N* at which long-run a preventive maintenance should undertaken (whichever comes first), average maintenance cost unit. numerical results are first), minimizing the average long-run maintenance costaction per time time unit.beObtained Obtained numerical results are discussed. discussed. first), minimizing the average long-run maintenance cost per time unit. Obtained numerical results are discussed. discussed. Keywords: Competing failures, degradation, preventive maintenance, shocks © 2015, IFAC (International Federation of Automatic Control) Hosting byreliability, Elsevier Ltd. All rights reserved. Keywords: Competing failures, degradation, preventive maintenance, reliability, shocks discussed. Keywords: Keywords: Competing Competing failures, failures, degradation, degradation, preventive preventive maintenance, maintenance, reliability, reliability, shocks shocks Keywords: Competing failures, degradation, preventive Systems maintenance, shocks have individual variations Systems havereliability, individual variations in in their their ability ability to to Systems have individual variations in their ability to Systems have individual variations in their ability to 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by Systems have individual variations their ability to 1. withstand shocks. Their to istypes: influenced by 1. INTRODUCTION INTRODUCTION withstand shocks. Their resistance to shocks shocks influenced by many factors which canresistance be classified classified inintwo internal many factors which can be in twois types: internal 1. INTRODUCTION withstand shocks. Their resistance to shocks is influenced by many factors which can be classified in two types: internal manyexternal. factors which can be classified in two internal and The factors due the The and external. The internal internal factors are are duetypes: the intrinsic intrinsic The failure failure of of various various systems systems and and equipment equipment are are generally generally manyexternal. factors which can be classified in two types: internal and external. The internal factors are due the intrinsic The failure of various systems and equipment are generally and The internal factors are due the intrinsic The failure of various systems and equipment are generally characteristics of the system (e.g. the quality of material, the due to competing failure modes. The first one is what is characteristics of the system (e.g. the quality of material, the due to competing competing failure modes. The first one one isgenerally what is is and external. The internal factors are due the intrinsic characteristics of the system (e.g. the quality of material, the The failure of various systems and equipment are due to failure modes. The first is what characteristics of the system (e.g. the quality of material, the due to competing failure modes. The first one is what is system structure and design, etc.). However, the external called catastrophic failure following some sudden external system structure and design, etc.). However, the external called catastrophic failure following some sudden external characteristics of the the However, quality of material, the system structure and design, etc.). the due to catastrophic competing failure modes. Thesome first one is external what is called failure following sudden system structure andsystem design,(e.g. the external external called catastrophic failureto some factors are generally related to the operation shocks due overheating or random voltage factors are generally related toetc.). the However, operation environment environment shocks due for for example example tofollowing overheating or sudden random external voltage system structure and design, etc.). However, the external factors are generally related to the operation environment called catastrophic failure following some sudden external shocks due for example to overheating or random voltage factors are generally related to the operation environment shocks due for example to overheating or random voltage (e.g. temperature, humidity, pressure, etc.). spikes or sudden and usage or (e.g. temperature, humidity, pressure, etc.). spikes or sudden and unexpected unexpected usage loads, loads, or accidental accidental factors are generally related tothethecumulative operation environment (e.g. temperature, humidity, pressure, etc.). shocks or due for hard example to overheating or random voltage spikes sudden and unexpected usage loads, or accidental (e.g. temperature, humidity, pressure, etc.). spikes or sudden and surfaces. unexpected usage loads, or is accidental In the literature, in the case of shock dropping onto The second one due to In the literature, in the case of the cumulative shock model, model, dropping onto hard surfaces. The second one is due to (e.g. temperature, humidity, pressure, etc.). In the literature, in the case of the cumulative shock model, spikes or sudden and unexpected usage loads, or accidental dropping onto hard surfaces. The second one is due to In the literature, in the assumed case of the cumulative shock model, dropping onto hard surfaces. The second onemodels is due are to the system is usually to be vulnerable to shocks cumulative physical deterioration. Shock the system is usually assumed to be vulnerable to shocks cumulative physical deterioration. Shock models are In the literature, infalls the assumed case of the cumulative shock model, the system is usually to be vulnerable to shocks dropping onto hard surfaces. The second onemodels is due are to cumulative physical deterioration. Shock the system is usually assumed to be vulnerable to shocks cumulative physical deterioration. Shock models are whose magnitude between 0 and a critical failure level. appropriate for modeling the reliability of systems subject to whose magnitude falls between 0 and a critical failure level. appropriate for modeling the reliability of systems subject to the system is usually assumed toand be to fact, shocks whose magnitude falls between 0 aavulnerable critical failure level. cumulative physical deterioration. Shock models are appropriate for modeling the reliability of systems subject to whose magnitude falls between 0 and critical failure level. appropriate for modeling the reliability of systems subject to Obviously, this assumption is not usually true. In in these kind of competing failure modes. Obviously, this assumption is not usually true. In fact, in these kind of competing failure modes. whose situations, magnitude falls between 0 and amagnitude critical level. Obviously, this assumption assumption is not not usually true.failure In fact, in appropriate modelingfailure the reliability these kind competing modes. Obviously, this is usually true. In fact, in these kind of offor competing failure modes. of systems subject to many shocks with small may have many situations, shocks with small magnitude may have Obviously, this assumption is not usually true. In fact, in many situations, shocks with small magnitude may have these kind of competing failure modes. studied by many many situations, with small magnitude may have negligible effect the process. Hence, it Shock negligible effect on onshocks the degradation degradation process. Hence, it would would Shock models models have have been been extensively extensively studied studied by by many many many situations, shocks with small magnitude may have negligible effect on the degradation process. Hence, it would Shock models have been extensively negligible effect on the degradation process. Hence, it would Shock models have been extensively studied by many be relevant to aa certain of magnitude, researchers (Chelbi et in to be relevanteffect to consider consider certain level level of shock shock magnitude, researchers (Chelbi et al., al., 2000)) 2000)) in order order to determine determine negligible on system the degradation process. Hence, it would be relevant to aa certain level of shock magnitude, Shock models have et extensively studied by These many researchers (Chelbi al., 2000)) in to be relevant to consider consider certain level oflevel, shock magnitude, researchers (Chelbi etbeen al., of 2000)) in order order to determine determine which we will call vulnerability below which mathematical expressions of systems reliability. which we will call system vulnerability level, below which mathematical expressions systems reliability. These be relevant to consider a certain level of shock magnitude, which we will call system vulnerability level, below which researchers (Chelbi et al., 2000)) in order to determine mathematical expressions of systems reliability. These which we will call system vulnerability level, below which mathematical expressions of systems reliability. These there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): whichis weno will callon system vulnerability level, below which there effect the system degradation process. This mathematical expressions ofcategories systems reliability. These models are classified into four (Peng et al., 2011): there is no effect on the system degradation process. This models are classified into four categories (Peng et al., 2011): vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude there is no effect oncan the system degradation process. This vulnerability level be estimated from shock and models are classified into four categories (Pengthe et al., 2011): (i) extreme shock model: failure occurs when magnitude vulnerability level can be estimated from shock and (i) extreme shock model: failure occurs when the magnitude deterioration data with classical statistical methods. of any shock exceeds some critical level (called resistance to deterioration data with classical statistical methods. of any shock exceeds some critical level (called resistance to vulnerabilitythelevel can beclassical estimated from shock and deterioration data with statistical methods. (i) any extreme shock model: occurs magnitude of shock exceeds some critical level (called resistance to deterioration with classical statistical methods. of any shock somefailure critical level when (calledthe resistance to Moreover, system vulnerability and its to shock) (Gut et (ii) shock model: Moreover, the data system vulnerability and its resistance resistance to shock) (Gut exceeds et al., al., 1999); 1999); (ii) cumulative cumulative shock model: deterioration data with classical statistical methods. Moreover, the system vulnerability and its resistance to of any shock exceeds some critical level (called resistance to shock) (Gut et al., 1999); (ii) cumulative shock model: Moreover, the system vulnerability and its resistance to shock) (Gut et al., 1999); (ii) cumulative shock model: shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks Moreover, system its resistance to shocks may the decrease as it itvulnerability gets older older and andand degrades. shock) occurs (Gut etwhen al., 1999); (ii)(Gut, cumulative shock model: failure the damage from shocks shocks may decrease as gets degrades. failure occurs when the cumulative cumulative damage(iii) from exceeds some threshold run shock exceeds some given given threshold (Gut, 1990); 1990); (iii) runshocks shock shocks may decrease as it gets older and degrades. failure occurs when the cumulative damage from shocks exceeds some given threshold (Gut, 1990); (iii) run shock exceeds some given threshold (Gut, 1990); (iii) run shock In model: occurs when is of In this this paper, paper, we we propose propose two two mixed mixed shock shock models models to to model model model: failure failure occursthreshold when there there is aaa run run(iii) of kkk shocks shocks In we two mixed models exceeds some given (Gut, is shock model: failure occurs when run of shocks model: failure occurs when there there is1990); a al., run2001); of run k and shocks In this this paper, paper, we propose propose two account mixed shock shock models to to model model systems reliability taking into the vulnerability level exceeding aa critical magnitude (Mallor et (iv) systems reliability taking into account the vulnerability level exceeding critical magnitude (Mallor et al., 2001); and (iv) In this paper, we propose twonoaccount mixed shock models to model systems reliability taking into the vulnerability level model: failure occurs when there isthe a al., run2001); of kbetween shocks exceeding aa critical magnitude (Mallor et and (iv) exceeding critical magnitude (Mallor et al., 2001); and (iv) systems reliability taking into account the vulnerability level below which shocks have effects on the degradation δ-shock model: failure occurs when time lag below which shocks have no effects on the degradation δ-shock model: failure occurs when the time lag between systems reliability taking intono account theon vulnerability level below which shocks have effects the degradation exceedingmodel: ashocks critical magnitude (Mallor et al., 2001); andet(iv) δ-shock failure occurs when the time lag between δ-shock model: failure occurs when the time lag between below which shocks have no effects on the degradation process. The first model combines the extreme shock and successive is shorter than a threshold δ (Wang al. process. The first first modelhave combines the extreme extreme shock and the the successive shocksfailure is shorter shorter thanwhen threshold (Wang et al. al. below which shocks no cumulative effects on degradation theshock degradation process. The model combines the and the δ-shock model: occurs the timeδδδ (Wang lag between successive shocks is than aaa threshold et successive shocks is shorter than threshold (Wang et al. process. The first model combines the extreme shock and due the cumulative shock models with 2001). cumulative shock models with cumulative degradation due 2001). process. Theshock first model combines theanextreme shock andfirst the cumulative models with cumulative degradation due successive shocks is shorter thanofa threshold δ (Wang et al. 2001). 2001). cumulative shock models with cumulative degradation due only to shocks. The second model is extension of the Furthermore, different mixtures these models have been only to shocks. shocks. Themodels secondwith model is an an extension extension of the the first first Furthermore, different mixtures mixtures of of these these models models have have been been cumulative shock cumulative due only The model is of 2001). Furthermore, different Furthermore, different mixtures of they theseare models havecalled been only to to shocks. The second second model is due an extension of the one with cumulative degradation to degradation shocks andfirst to investigated by several researchers; generally one with cumulative degradation due to shocks and to investigated by several researchers; they are generally called only to shocks. The second model is an extension of the first one with cumulative degradation due to shocks and to Furthermore, different mixtures of these models have been investigated by several researchers; they are generally called investigated by several researchers; they are generally called one with cumulative degradation due to shocks and to internal factors. factors. mixed shock models. Shanthikumar et al. (1983) proposed aa internal mixed shock models. Shanthikumar et al. (1983) proposed one with cumulative degradation due to shocks and to internal factors. investigated by several researchers; they are generally called mixed shock models. Shanthikumar et al. (1983) proposed a mixed shock models. Shanthikumar et al. (1983) proposed a internal factors. general model which the shock model general modelmodels. which combines combines the extreme extreme shock proposed model and and internal mixed shock Shanthikumar etbased al. (1983) general model which combines the shock model and general model which combines the extreme extreme shock model anda We also alsofactors. propose aa preventive preventive maintenance maintenance (PM) (PM) policy policy for for δ-shock model. This combination is on the fact that We propose δ-shock model. This combination is based on the fact that We also propose aa preventive maintenance (PM) policy for general model which combines the extreme shock model and δ-shock model. This combination is based on the fact that δ-shock model. This combination is based on the fact that We also propose preventive maintenance (PM) policy for systems whose reliability is modeled according to the second the magnitude of the k-th shock and the time interval systems whose reliability is modeled according to the second the magnitude of thecombination k-th shock shockis and and theontime time interval We alsoshock propose a preventive maintenance (PM) policy for systems whose reliability is modeled according to the second δ-shock model. Thisthe basedthe the fact that the magnitude of k-th interval the magnitude of the k-th shock and the time interval systems whose reliability is modeled according to the second mixed model involving degradation due to internal between consecutive shocks are correlated with each other. mixed shock model involving degradation due internal between consecutive shocks are correlated with each each other. systems whose reliability is modeled according to to the second mixed shock model involving degradation due to internal the magnitude of the k-thare shock and the interval between consecutive shocks correlated with between consecutive shocks are correlated with time each other. other. mixed shock model involving degradation due to internal and external factors. System is replaced replaced following failure or and external factors. System is following failure or mixed shock model involving degradation due to internal and external factors. System is replaced following failure or between consecutive shocks are correlated with each other. and external factors. System is replaced following failure or 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015 IFAC 2238 and external factors. System is replaced following failure or ** University **University
Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 2238Control. Copyright © 2238 Copyright © 2015 2015 IFAC IFAC 2238 10.1016/j.ifacol.2015.06.409 Copyright © 2015 IFAC 2238
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preventively at an age T or after having been subject to N shocks. A mathematical model is developed to determine the optimal age T* and the number of shocks N*, which minimize the average long-run maintenance cost per time unit.
(i.e. the time from any time t to the next shock is independent of time t). Shocks occur with a stationary rate 𝜆𝜆. Let N(t) denote the number of random shocks until time t. Then the probability Hi(t) that shocks occur exactly i times in time interval (0, t] is given by: H i t P N t i
In the literature, there are many maintenance policies based on shock models. Wang et al. (2001) consider a system whose failure is induced by the δ-shock model. They determine the optimal failure number N* such that the longrun average cost per time unit is minimized. This work has been extended by Wang et al. (2005) considering a mixed shock model (i.e. extreme shock model and δ-shock model). They consider a repairable system subject to two types of failure. The first one is the result of the inter-arrival time between consecutive shocks being shorter than a given threshold value. The second one is based on the magnitude of a single shock that is higher than a given threshold level value. Nakagawa (1976) considers the cumulative shock model. He proposes a replacement policy according to which an item is replaced at failure or when a certain level of cumulative damage is reached, whichever occurs first. A mathematical framework is developed to obtain the optimal level of damage which minimizes the long-run cost per unit of time. Nakagawa et al. (1989) propose a periodic replacement policy with minimal repair at failure considering the cumulative shock model. They obtain the optimal solution for the time T*, number of shocks N*, and damage level Z* at which preventive replacement should be performed. Qian et al. (2005) proposed a maintenance policy considering an extended cumulative shock model with shocks occurring according to a non-homogeneous Poisson process. The decision variables are the failures number N and the age T.
t
i
(1)
e t
i! (B) Shocks follow a non-homogeneous Poisson Process (NHPP) when shocks arrival is not stationary. In such situations, if t stands for the shocks occurrence rate, then the probability Hi(t) that shocks occur exactly i times in time interval (0, t] is given by:
t e i
Hi t
t
i!
(2)
t
Where t u du 0
If shocks occur at times T1, T2, …, according to NHPP, then the probability that more than (i+1) shocks occur during (0,t] is:
P T i 1 t H i u u du t
0
H t
(3)
j
j i 1
In this section, we consider a system subject to shocks which occur at times Tj (j = 1, 2,…). The shock magnitudes are denoted by Wi. They are assumed to be random variables, sindependent, and identically distributed with a common distribution GW where GW(0)=0. In order to express the system reliability function, the mixed shock model that is considered here is based on the one which combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks (Gut, 2001)). Failures occur following an extreme shock whose magnitude is higher that a given critical level U (extreme shock model), or whenever cumulative damage exceeds a given degradation level threshold. The total degradation is assumed to be accumulating through shocks whose magnitude is between zero and U (cumulative shock model). In fact, this assumption is not always true. In many situations, shocks with small magnitude (minor shocks) may have no effect on the degradation process. Hence, we introduce a vulnerability level, L, below which shocks have no effect on the system degradation process; they will be called minor shocks. Both levels L and U are either independent of time t or decreasing with respect to operating time (or number of shocks). Keeping them constant means that the system can recover from the consequences of a previous shock (i.e. the mean inter-arrival time of shocks is much larger than the mean time of recovery). In case extreme shock and cumulative shock provoke an increase in the probability of system failure, L(t) and U(t) are considered as decreasing with time (or number of shocks). This is usually the case for many machines and equipment. In this work, we consider the general case (L(t) and U(t): constant or decreasing functions).
The remainder of this paper is organized as follows. Section 2 is dedicated to the development of a generalized reliability expression using a mixed shock model which combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks. In section 3, we develop the generalized expression of system reliability in the case of a mixed degradation and shock model considering cumulative degradation due to shocks and to internal factors. Section 4 is devoted to the definition and mathematical modeling of a preventive maintenance policy. The latter suggests performing PM actions at shock number N or at age T, whichever occurs first. A numerical example is discussed in Section 5. Finally, some comments and conclusions are drawn in the last section. 2.
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MODEL 1: GENERALIZED MIXED SHOCK MODEL
Generally shocks occurrences over time are modeled according to a Poisson process. We distinguish two kinds of Poisson processes: (A) Shocks follow a homogeneous Poisson process (HPP) when the times between successive shocks are distributed exponentially and have a memoryless property. In other words, shocks are generated randomly and uniformly in time 2239
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Our proposed generalized mixed shock model will combine three major types of shocks - namely extreme shock, cumulative shock and minor shock. System failure occurs due to an extreme shock whose magnitude exceeds some given critical level U(t), whereas cumulative shock causes a cumulative damage to system only in case the shock magnitude is higher than the vulnerability level L(t) and lower than the critical level U(t). Failures due to cumulative shocks occur whenever the cumulative damage exceeds a certain threshold level D. Minor shocks (whose magnitude is lower than the vulnerability level L(t)) do not cause any damage to the system. The proposed generalized mixed shock model is summarized as follows considering a given shock of magnitude Wi occurring at time Ti:
Yielding:
R1 D , t H 0 t H i t G si D
Stieltjes convolution of any distribution Gs(x) with itself; Gs(0)(x)=1 for x≥0. 3. MODEL 2: GENERALIZED MIXED DEGRADATION AND SHOCK MODEL In this section, we extend the generalized mixed shock model presented in section 2 considering that cumulative degradation is due not only to shocks but also to internal factors. Thus, the overall damage of the system is cumulative. It is caused by (i) internal degradation through a random process function, X d t which represents the
degradation process causing additional damage X s Ti
deterioration of the system corresponding to working time and (ii) external random shocks inducing cumulative damage expressed by X s Ti , i 0,1, 2,... .
Wi L Ti the shock has no effect on the system
degradation process (minor shock). Events E.1, E.2 and E.3 are mutually exclusive.
Failure occurs whenever the cumulative degradation process exceeds a given threshold level D or shock magnitude exceeds a critical level U(t), whichever occurs first. The cumulative degradation is expressed as: N t
X t X d t X s T j I T , L ,U
i 0,1, 2,... (4)
j 0
R 2 D ,t P X t D
N t P X d t X s T j I T , L ,U D | N t i j i 0 j 0
L t W i U t U t
P N t i
dGWi z
P X d t D ; I T i , L ,U 0; i 0
GWi U t GWi L t
W 1 L t ,W 2 L t ,...,W i L t | N t i P N t i
(5)
N t P X d t X s T j D ; i 1 j 0
The cumulative damage due to random shocks by time t is given as: N t
X t X s Ti I Ti , L,U
(9)
j
Therefore, the system reliability at time t for a given threshold level D of cumulative damage is given by:
Suppose i shocks occur before time t, the probability that the intensities of each of these shocks are between L(t) and U(t) is given by: PU , L t P L t W 1 U t , L t W 2 U t ,...,
L t
(8)
of X s t and Gs(j)(x) (j = 1, 2, · · · ) denotes the j-fold
E.2 L Ti Wi U Ti the shock occurrence accelerates the
dGWi z
where Gs ( x) is the Cumulative distribution function (Cdf)
system
We consider the following binary variable: 1 if L T i W i U T i I Ti , L ,U 0 if W i L T i Where T0 0 and T0 , L ,U =0
L t
i 1
E.1 U Ti Wi the shock causes immediate failure of the
E.3
U t
L t W 1 U t , L t W 2 U t ,..., L t W i U t | N t i P N t i
(6)
i 0
Hence, the survival distribution of the system can be expressed as follows: R1 D , t P X t D
(10)
N t
which yields the following expression: P X s T j I T , L ,U D | N t i j i 0 j 0 R 2 D , t H 0 t Fd D , t H i t Fd D ,t GWi L t ×P N t =i i 1
D U t N t H i t Fd D u , t dG si u dGWi z P X s T j D ; I T , L ,U 1| N t i L 0 t j i 1 i 0 j 0 (11) ×P N t =i Where Fd ( x, t ) is the Cdf of X d t (7)
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It is interesting to note that the two proposed reliability models are a generalization of several past studies. For instance in case L(t)=0 and U(t)=U (constant and finite): models 1 and 2 are reduced to two types of dependent failure processes i.e. cumulative shock and extreme shock model. Hence, we obtain the following results as obtained by (Pen et al. (2001):
R1 D , t H 0 t H i t G s
i
D GWi U
Our objective is to determine simultaneously the optimal age T*, as well as the number of shocks N* to perform PM such as the average long-run maintenance cost rate is minimum. Using classical renewal arguments, the total average maintenance cost per time unit can be expressed over a renewal cycle S. In fact, as previously stated, the system is considered to be as good as new after all maintenance actions.
(12)
i 1
Hence, the expression of the long-run average maintenance cost per time unit is given by:
R 2 D , t H 0 t Fd D , t
H i t Fd D u , t dG si u GWi U D
EC T , N
0
i 1
(13) 4.
2167
E CTotal T , N
(14)
E TS T , N
Where E CTotal represents the expected total maintenance
MAINTENANCE POLICY
cost incurred within a renewal cycle, and E TS is the mean operating time (average duration of a renewal cycle). The following analysis will lead to the expression of the average long-run maintenance cost per time unit.
In this section, we propose a preventive maintenance policy for a system subject to the generalized mixed degradation and shock model (model 2 presented in section 3) considering three types of shocks. Minor Shocks with a
relatively low magnitude ( Wi L Ti ) have no effect on
4.1 Expected total cost within a cycle
the system, shocks with a high level of magnitude (
The expected total maintenance cost during a cycle E CTotal can be expressed as follows:
U Ti Wi ) (extreme shocks) cause immediate failure, and
shocks
with
an
intermediate
magnitude
level
(
E CTotal (T , N ) CT PT (N ,T ) C N PN (N ,T ) C D PD (N ,T ) CU PU (N ,T )
L Ti Wi U Ti ) accelerate the degradation process by
some amount X s Ti . The cumulative damage is due to
internal degradation and external random shocks. The considered system experiences two competing dependent failure modes: type-I failure and type-II failure. Type-I failure occurs when cumulative degradation exceeds a given threshold level D. Type-II failure occurs due an extreme shock. The system is replaced preventively before failure at an age T (type-I PM action), or after having been subject to a number of shocks N (type-II PM action), whichever occurs first.
(15)
The four terms of the total maintenance cost correspond respectively to type-I PM, type-II PM, type-I CM and type-II CM costs during a renewal cycle. Analytical expressions of each of these different components are developed below. -
The probability PT of having a type-I PM (at age T) is
expressed as follows: N T PT T , N P X d T X s T i T i , L ,U D ; i 0
W 1 U ,W 2 U ,...,W i U ; N T N
Hence, the proposed maintenance policy preconizes the following maintenance actions:
N 1
H i T
- Type-I and type-II PM actions: carried out at age T and at
i 1
shock number N, respectively;
D
0
Fd D u ,T dG si u dGWi z U
L
N 1
H i T Fd D ,T GWi L
- Type-I
corrective maintenance CM action: performed whenever the total damage exceeds a threshold level D.
i 0
(16)
- Type-II
CM action: carried out in case one shock’s magnitude exceeds the critical level U.
-
The probability PN of having a type-II PM (after N
successive shocks) is given by: N PN T , N P X d T X s T i T i , L ,U D ; i 0
Working assumptions: - Random shocks occur according to a Poisson process. - Failures are self-announced and are immediately fixed. - All maintenance actions renew or bring back the system to a state as good as new. - Durations of maintenance actions are negligible. - The costs of Type-I PM (CT), Type-II PM (CN), Type-I CM (CD) and Type-II CM (CU) are constant and known. - The critical level, the vulnerability level and the degradation threshold level are constant.
W 1 U ,W 2 U ,...,W N U ; T N T T
D
U
H N 1 t t dt Fd D u ,T dG s N u dGWN z 0
0
T
L
H N 1 t t dt Fd D ,T G 0
N W
L (17)
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5.
The probability PD of having a type-I CM (an exceeding of threshold level D of cumulative degradation) is given by. j PD T , N P X d T X s T i T i , L ,U D ; i 0 -
NUMERICAL EXAMPLE
In order to discuss the developed reliability functions for both models 1 and 2 and illustrate the application of the maintenance strategy, we consider for instance MicroElectro-Mechanical Systems –MEMS. These systems are j 1 small integrated devices that combine micro-electrical and X d T X s T i T i , L ,U D ; micro-mechanical components and permit the inclusion of i 0 mechanical elements such as gears, diaphragms, and springs W 1 U ,W 2 U ,...,W j 1 U ; j 1 N ;T j 1 T to integrated circuits. We consider a system that contains one microengine. Such a system was studied by Tanner et al. 1 Fd D ,T (2003). The microengine consists of gears connected N 1 T GWi L GW U mechanically to orthogonal linear comb drives and springs. 0 H i t t dt Fd D ,T G i 1 L Rotation of gears is ensured by linear displacement of the i 0 W comb drives. The microengine is assumed to be subject to D F D u ,T dG i u two competing failure processes; wear degradation due to s 0 d aging and debris from shock loads, and spring fracture due to U i a large shock load (Jiang et al. (2012)). N 1 T L dGW z GW U In the following numerical example, time t will be expressed H i t t dt D 0 i 0 Fd D u ,T dG s i 1 u in terms of number of revolutions. Suppose that wear 0 degradation due to aging increases linearly, Xd(t)=x0+βt, U dG i 1 z where β is a normally distributed random variable, β∼N(𝜇𝜇β, W L 𝜎𝜎β2), and wear degradation due to debris from shock loads (18) follows normal distribution Xs∼N(𝜇𝜇s, 𝜎𝜎s2) (Peng et al., 2011). - The probability PU of having type-II CM (extreme We assume also that shocks magnitudes are distributed shock) is given by: exponentially GW t 1 exp t , where θ = 1.2GPa, and j shocks occur according to a Poisson process with stationary PU T , N P X d T X s T i T i , L ,U D , rate λ. i 0 The following input parameters are given below in Table1. W 1 U ,W 2 U ,...,W j U ,W j 1 U ; They are similar to those used by Peng et al. (2011). The obtained results are presented in the next subsections. j 1 N ; T j 1 T N 1 T
H i t t dt Fd D u ,T dG s i u i 1
D
0
Table 1. Input data
0
U
dG L
i W
z U dGW y
H i t t dt Fd D ,T G
i W
0
1.5
L U dGW y
E T S R t , N 1dt T
0
i 1
T
0
N 1 T
H i t Fd D u , t dG s D
0
0
0
𝜎𝜎β
𝜇𝜇s
(𝜇𝜇m3) (𝜇𝜇m3)
8.4823 6.0016 10-9 10-10
10-4
𝜎𝜎s
λ
2 10-5
2.5
(𝜇𝜇m3) (10-5)
U
u dt L dGWi z (20)
R1(D,t)
L=1.5 = U
H i t Fd D , t dtG i 0
i
12.5 10-4
𝜇𝜇β
(𝜇𝜇m3)
Using the input data given above, we computed the reliability functions corresponding to both models 1 and 2 for different values of the system vulnerability level L. These survival functions are presented in figures 3 and 4.
The mean operating time between successive maintenance actions can be expressed as follows: N 1
x0
5.1 Reliability functions behavior relative to the system vulnerability level
4.2 Average duration of a renewal cycle:
D
U
(GPa) (𝜇𝜇m3) (𝜇𝜇m3)
N 1 T i 0
(19)
i W
L
L=1.0 L=0.5
Hence, the average long-run cost rate function CTotal (T , N ) can be obtained by combining equations (16) to (20). The resulting expression is a function of the decision variables which are the age T and the number of shocks N . A numerical procedure is designed to obtain the optimal preventive maintenance policy. The procedure searches for different values of T∊ (0, Tmax] and N∊[1, Nmax] to determine the optimal age T* and the number of shocks N* which minimize the average long-run maintenance cost rate.
L=0.0
t Fig. 3: Reliability behavior in case of model 1 2242
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probability of occurrence of failure, the system is maintained preventively at a certain age T* or after a certain number of shocks N* (whichever occurs first). A mathematical expression of the expected total maintenance cost rate has been developed, and a numerical procedure has been used to determine simultaneously the optimal values of the decision variables T* and N*. The study of run shock model and δshock models when the system vulnerability level is taken into account represent one of the major possible extensions of this work. It is currently under consideration by the authors.
R2(D,t) L=1.5 = U L=1.0 L=0.5 L=0.0
REFERENCES
t
Chelbi, A. and Ait-Kadi, D. (2000) Generalized Inspection Strategy for Randomly Failing Systems Subjected to Random Shocks, International Journal of Production Economics, Vol. 64, pp-379384. Jiang L., Feng Q. and Coit D.W. (2012) Reliability and Maintenance Modeling for Dependent Competing Failure Processes With Shifting Failure Thresholds, IEEE Transactions On Reliability, vol. 61, no. 4. Gut, A. (1990) Cumulative shock models, Advances in Applied Probability 22:504–507. Gut, A. (2001) Mixed shock models, Bernoulli 7(3):541– 555. Gut, A. and Husler, J. (1999) Extreme shock models, Extremes 2, 293-305. Li, ZH. and Kong, XB. (2007) Life behavior of d-shock model, Statistics & Probability Letters 77:577–587. Mallor, F. and Omey, E. (2001) Shocks, runs and random sums, Journal of Applied Probability 38, 438-448. Nakagawa, T. (1976) On a replacement problem of a cumulative damage model. Operational Research Quarterly, 895-900. Nakagawa, T. and M. Kijima (1989). Replacement policies for a cumulative damage model with minimal repair at failure. IEEE Trans. Reliability, 13, 581-584. Peng, H., Feng, Q. M., and Coit D. W. (2011) Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes, IIE Transactions, vol. 43, pp. 12-22. Qian, CH, Ito K and Nakagawa T (2005). Optimal preventive maintenance policies for a shock model with given damage level. Journal of Quality in Maintenance Engineering 11:216–227. Shanthikumar, JG. and Sumita U. (1983) General shock models associated with correlated renewal sequences, Journal of Applied Probability 20:600–614. Tanner, D.M. and Dugger, M.T. (2003) Wear mechanisms in a reliability methodology, Proceedings of the Society of Photo-optical Instrumentation Engineers, 4980, 22–40. Wang, G.J. and Zhang Y.L. (2001) δ-shock model and its optimal replacement policy, Journal of Southeast University, 31, pp. 121–124. Wang, G. J. and Zhan, Y. L. (2005) A shock model with two-type failures and optimal replacement policy, International Journal of Systems Science, 36:4, 209214.
Fig. 4: Reliability behavior in case of model 2 5.2 Optimal PM policy In this subsection, we assume that the vulnerability level is fixed and constant, L=1.0 GPa. The costs of type-I PM, typeII PM, type-I CM and type-II CM have been arbitrarily chosen as: 20$, 20$, 190$ and 250$, respectively. We applied the numerical procedure varying T and N as follows T∊(0,106] and N∊[1, 10]. The obtained optimal solution is presented in Table 2. Table 2. The obtained optimal solution of preventive maintenance policy T* E[TS]* EC* N* PT PN PD PU (revolutions) (revolutions) ($/cycle) 9 104
4
0.61 0.09 0.005 0.30
7.3104
2169
90
Hence, the maintenance crew has to replace the system preventively before failure after T*=9x104 revolutions (typeI PM action), or after having been subject to N*=4 shocks (type-II PM action), whichever occurs first. The corresponding total average maintenance cost is equal to 90$/cycle with an average cycle duration of 7.3x104 revolutions. 6. CONCLUSIONS In this paper, we proposed a generalization of two mixed shock models to express systems reliability taking into account, what we called the system vulnerability level below which shocks have no effects on the degradation process. The first model combines the extreme shock and the cumulative shock models with cumulative degradation due only to shocks. The second model is an extension of the first one with cumulative degradation due to shocks and to internal factors. The developed expressions represent a generalization of some previously obtained results in the literature. We also proposed a preventive maintenance policy considering for systems subject to the generalized mixed degradation and shock model. The system is subject to two competing dependent failure modes due to either extreme shock, or the accumulation of damage. To lower the 2243