- Email: [email protected]
Imperfect Inspection Policy for Systems with Multiple Correlated Degradation Processes
IFAC Conference Manufacturing Modelling, Management and on Control IFAC Conference Manufacturing Modelling, Management and on Control June 28-30, 2016. Troyes, France Modelling, IFAC on Manufacturing IFAC Conference Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com June 28-30, 2016. Troyes, Management and and Control Control France Management June 28-30, 2016. Troyes, France June 28-30, 2016. Troyes, France June 28-30, 2016. Troyes, France
ScienceDirect
IFAC-PapersOnLine 49-12 (2016) 1377–1382 IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE CORRELATED DEGRADATION PROCESSES IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH IMPERFECTCORRELATED INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE MULTIPLE DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo*
Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo* Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo* Bin Liu*, Zhao*, Ruey-Huei Yeh**, Bin Engineering Liu*, Xiujie Xiujieand Zhao*, Ruey-Huei Yeh**, Way Way Kuo* * Department of Systems Engineering Management, CityKuo* University of Hong Kong, * Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) * Department of Systems Engineering and Engineering Management, City University of Hong Kong, *** Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) * Department of Systems Engineering and Engineering Management, City University of Hong Kong, Department of(e-mail: Industrial Management, National Taiwan University of Science [email protected]) Technology, Kowloon, Hong Kong [email protected]; [email protected]; Kowloon, Hong Kong [email protected]; [email protected]; ** Department of(e-mail: Industrial Management, National Taiwan University of Science [email protected]) Technology, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) Taipei, Taiwan (email: [email protected])} ** Management, National Taiwan ** Department Department of of Industrial Industrial Management, National Taiwan University University of of Science Science and and Technology, Technology, Taipei, Taiwan (email: [email protected])} ** Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan (email: [email protected])} Taipei, Taipei, Taiwan Taiwan (email: (email: [email protected])} [email protected])} Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated Abstract: This paper develops an imperfect inspection policy for systems subject toWiener multiple correlated degradation processes. The degradation processes are characterized by a multivariate process, and Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated degradation processes. The degradation processes are characterized by a multivariate Wiener process, and Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated the dependency among the processes is described by a covariance matrix. A failure occurs when any of degradation processes. The degradation processes are characterized by aa multivariate Wiener process, and degradation processes. The are characterized by multivariate Wiener process, the dependency among the degradation processes is processes described byAs a covariance matrix. A failureare occurs when anyand of degradation processes. The degradation processes are characterized by a multivariate Wiener process, and the degradation level exceeds a specific threshold. degradation-based failures usually dormant, the dependency among the processes is described by aa covariance matrix. A failure occurs when any of the dependency among processes is by matrix. A failure occurs when any degradation level exceeds athe specific threshold. As degradation-based failures are usually dormant, the dependency among the processes is described described byAs a covariance covariance matrix.The A failure occurs when any of of inspection is required tothe detect failures and prevent potential losses. inspection is imperfect, in the degradation level exceeds a specific threshold. degradation-based failures are usually dormant, the degradation level exceeds aathe specific threshold. As degradation-based failures are usually dormant, inspection isthat required to detect failures and prevent potential Optimal losses. The inspection is imperfect, in the degradation level exceeds specific threshold. As degradation-based failures are usually dormant, such a way a failure may not be discovered at an inspection. inspection interval is obtained inspection is required to detect the failures and potential losses. The inspection is in inspection isthat required to run detect therate. failures and prevent prevent potential Optimal losses. The inspection is imperfect, imperfect, in such a wayis a failure maycost not befailures discovered at an inspection. inspection interval is obtained inspection required to detect the and prevent potential losses. The inspection is imperfect, in by minimizing the long In addition, theinspection. properties of the optimal inspection interval are such a way that a failure may not be discovered at an Optimal inspection interval is obtained such aa way that aa failure may not be discovered at an Optimal inspection interval is obtained by minimizing the long run cost rate. Inofaddition, theinspection. properties of the optimal inspection interval are such way that failure may not be discovered at an inspection. Optimal inspection interval is obtained studied and the upper and lower bounds the optimum value are achieved. An example of fatigue crack by minimizing the long run cost rate. In the properties of optimal inspection interval are by minimizing the long run costbounds rate. Inofaddition, addition, the properties of the the optimal inspection interval are studied and the upper andillustrate lower the optimum value are achieved. An example of fatigue crack by minimizing run cost rate. In addition, properties of the inspection interval are development is the usedlong to and evaluate the the inspection policy. Theoptimal results imply of that inspection studied and the upper and lower bounds of the optimum value are achieved. An example fatigue crack studied and the and lower bounds of the optimum value are achieved. An example of fatigue crack development is upper used significantly to illustrate and evaluate the operating inspection policy. The results imply that inspection studied and the upper and lower bounds of the optimum value are achieved. An example of fatigue crack accuracy contributes to reducing the cost and more efforts should be made to development is used to illustrate and evaluate the inspection policy. The results imply that inspection development is used to illustrate and evaluate the inspection policy. imply that accuracy contributes significantly to reducing operating cost and The moreresults efforts should beinspection made to development is used to illustrate and evaluate the inspection policy. The results imply that inspection improve the inspection quality. accuracy contributes significantly to reducing the operating cost and more efforts should be made to accuracy contributes to the cost and efforts should be improve the inspectionsignificantly quality. accuracy contributes significantly to reducing reducing the operating operating cost hidden and more more efforts shouldanalysis be made made to to improve the inspection quality. Keywords: Imperfect inspection; correlated degradation processes; failure; reliability © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. improve the inspection quality. improve theImperfect inspectioninspection; quality. correlated degradation processes; hidden failure; reliability analysis Keywords: Keywords: processes; Keywords: Imperfect Imperfect inspection; inspection; correlated correlated degradation degradation processes; hidden hidden failure; failure; reliability reliability analysis analysis Keywords: Imperfect inspection; correlated degradation processes; hidden failure; reliability analysis size of any crack exceeds a specific threshold. The cracks 1. INTRODUCTION size of correlated any crack inexceeds a specific cracks may be such a way that allthreshold. the cracksThe are subject size of any crack exceeds aa specific threshold. The cracks 1. INTRODUCTION size of any crack specific The cracks may be correlated inexceeds such awheel way that allthreshold. theand cracks are subject 1. INTRODUCTION size of any crack exceeds a specific threshold. The cracks to common shocks, e.g., fraction environmental Nowadays, maintenance has gained increasing attention as 1. INTRODUCTION may be correlated in such a way that all the cracks are subject 1. INTRODUCTION may be correlated in such a way that all the cracks are subject to common shocks, e.g., wheel fraction and environmental Nowadays, maintenance hasimprove gained increasing attention as may be correlated in such a way that all the cracks are subject corrosion. Maintenance policy on multiple correlated maintenance operations can production quality and to common shocks, e.g., wheel fraction and environmental Nowadays, maintenance has gained increasing attention as to common shocks, wheel fraction and Nowadays, maintenance has gained as corrosion. Maintenance policy onresearch. multiple correlated maintenance operations can improve production quality and to common processes shocks, e.g., e.g., wheel fraction and environmental environmental requires further Nowadays, maintenance has gained increasing increasing attention as degradation reduce losses caused by unexpected failures. attention Traditional corrosion. Maintenance policy on multiple correlated maintenance operations can improve production quality and corrosion. Maintenance policy on multiple maintenance operations can improve production quality and degradation processes requires further research. reduce losses caused by unexpected failures. Traditional corrosion. processes Maintenance policy onresearch. multiple correlated correlated operations can improvebased production quality and degradation maintenance actions areby performed on failure models, requires further reduce losses caused unexpected failures. Traditional Failure caused by degradation is referred to as “hidden degradation processes requires further research. reduce losses caused by unexpected failures. Traditional maintenance actions are performed based on In failure models, degradation processes requires further research. reduce losses caused by unexpected failures. Traditional which are constructed by use of failure data. recent years, Failure or caused by degradation isal, referred to asis “hidden maintenance actions are performed based on failure models, failure” “soft failure” (Peng et 2010), which usually maintenance actions are based on failure models, caused by degradation is referred to as “hidden which aredue constructed byperformed use of failure data. recent years, Failure maintenance actions are performed based on In failure models, however, to improved quality and reliability of products, Failure caused by degradation to as failure” or “soft failure” (Peng et is al,bereferred 2010), which is “hidden usually which are constructed by use of failure data. In recent years, Failure caused by degradation is referred to as “hidden dormant to engineers and can only revealed by inspection which constructed by of data. years, or “soft failure” (Peng et al, 2010), which is usually however, due to improved quality and reliability of which are constructed by use use of failure failure data. In InIt recent recent years, failure” failure are data are becoming harder to obtain. is products, therefore failure” or “soft failure” (Peng et al, 2010), which is usually dormant to engineers and can only be revealed by inspection however, due to improved quality and reliability of products, failure” or “soft failure” (Peng et al,bearticles 2010), which is that usually or testing. Majority ofand the existing assume the however, due to improved improved quality and reliability of products, dormant to engineers can only revealed by inspection failure data are becoming harder and to obtain. It of is on therefore however, due to quality reliability products, difficult to implement maintenance activities based failure dormant to engineers and can only be revealed by inspection or testing. Majority of the existing articles assume that the failure data are becoming harder to obtain. It is therefore dormant to engineers and can only be revealed by inspection inspection is perfect. However, in reality, inspection is failure data data are Xie, becoming harder activities to obtain. obtain.based It is is on therefore testing. Majority of the existing articles assume that the difficult to implement maintenance failure or failure are becoming harder to It therefore models (Ye and 2015). or testing. Majority of the existing articles assume that the inspection is perfect. However, in reality, inspection is difficult to implement maintenance activities based on failure or testing. Majority of the existing articles assume that the usually less than perfect; the hidden failure cannot always be difficult to implement maintenance activities based on failure is perfect. However, in reality, inspection is models and Xie, 2015). difficult(Ye to implement maintenance activities based on failure inspection inspection is perfect. However, in reality, inspection is usually less than perfect; the hidden failure cannot always be models (Ye and Xie, 2015). inspection is perfect. However, in reality, inspection is spotted less by an inspection (Berrade etfailure al, 2013). During one As an alternative, degradation models provide opportunities usually models (Ye and Xie, 2015). than perfect; the hidden cannot always be models (Ye and Xie, 2015). usually less than perfect; the hidden failure cannot always be spotted by an inspection (Berrade et al, 2013). During one As an alternative, degradation models provide opportunities usually less than perfect; the hidden failure cannot always be inspection, the hidden failure can only be detected with some for an reliability analysis and maintenance decisionopportunities making for spotted by an inspection (Berrade et al, 2013). During one As alternative, degradation models provide spotted probability. by the an inspection inspection (Berrade et al, al, 2013). During During one As an an alternative, degradation modelsDegradation provide opportunities inspection, hidden failure can only be detected with some for reliability analysis and decisionopportunities making for spotted by an (Berrade et 2013). one certain As alternative, degradation models provide highly reliable products andmaintenance systems. models are inspection, the hidden failure can only be detected with some for reliability analysis and maintenance decision making for inspection, the hidden failure can only be detected with some for reliability analysis and decision making for certain probability. highly reliable products andmaintenance systems. Degradation models are inspection, the hidden failure can only be detected with some for reliability analysis and maintenance decision making for constructed by use of degradation measurements, which can certain probability. highly reliable products and systems. Degradation models are No existing maintenance strategy has covered the joint effect certain probability. highly reliable products and Degradation are constructed by use operation of degradation measurements, which certain probability. highly reliable products and systems. systems. Degradation models are be recorded during of a product or systemmodels (Ye etcan al, No existing maintenance processes strategy has the joint effect constructed by use of degradation measurements, which can of multiple degradation andcovered imperfect inspection. constructed by use of degradation measurements, which can No existing maintenance strategy has covered the effect be recorded during operation of a product or system (Ye et al, constructed by use of degradation measurements, which can 2012). Another advantage ofofdegradation model lies in its of Nomultiple existing maintenance strategy has covered the joint joint effect degradation processes andcovered imperfect inspection. be recorded during operation a product or system (Ye et al, No existing maintenance strategy has the joint effect Our research aims to fulfill the gap and contribute to the be recorded during operation of a product or system (Ye et al, of multiple degradation processes and imperfect inspection. 2012). Another advantage of degradation model lies in its be recorded operation ofdegradation a product system (Yein et al, ability to during characterize physical failureormodel mechanism. In of multiple degradation processes and imperfect inspection. Our researchof aims to fulfill the gap and contribute to the 2012). Another advantage of lies its of multiple degradation processes and imperfect inspection. knowledge maintenance concerning multiple dependent 2012). Another advantage of degradation model lies research aims to fulfill the gap and contribute to the ability todegradation-based characterize failure mechanism. In 2012). Another advantagephysical of model degradation model lies in in its its addition, provides flexibility to Our Our research aims to the contribute to knowledge maintenance dependent ability to characterize physical failure mechanism. In Our researchof aims to fulfill fulfill the gap gap aand andmultiple contribute to the the degradation processes by concerning adding realistic factor of ability to characterize physical failure mechanism. In knowledge of maintenance concerning multiple dependent addition, degradation-based model provides flexibility to ability todegradation-based characterize failure mechanism. In investigate the impact ofphysical environmental variation on the knowledge of maintenance degradation processes by concerning adding a multiple realistic dependent factor of addition, model provides flexibility to knowledge of maintenance concerning multiple dependent imperfect inspection. addition, degradation-based model provides provides flexibility to degradation processes by adding a realistic factor of investigate the impact of environmental variation on the addition, degradation-based model flexibility to deteriorating process. degradation processes imperfect inspection. investigate the impact of environmental variation on the degradation processes by by adding adding aa realistic realistic factor factor of of investigate impact of environmental variation on the imperfect inspection. deterioratingthe process. investigate the impact of environmental variation on the In this study, we develop an imperfect inspection policy for imperfect inspection. deteriorating process. imperfect inspection. An implicit assumption of most existing works on deteriorating process. In this study, imperfectdegradation inspection processes. policy for deteriorating subjectwe to develop multiplean correlated In this study, we develop an imperfect inspection policy for An implicit process. assumption of ismost existing works on system degradation-based maintenance that only one degradation In this study, develop imperfect inspection policy system subjectwe todegradation multiplean correlated degradation processes. An implicit assumption of most existing works on In this study, we develop an imperfect inspection policyasfor fora The dependent processes are modeled An implicit assumption of most existing works on system subject to multiple correlated degradation processes. degradation-based maintenance is that only one degradation An implicit of system. most existing works on system process imposesassumption impact on the However, in reality, subject to multiple correlated degradation processes. The dependent degradation processes are modeled as a degradation-based maintenance is that only one degradation system subject to multiple correlated degradation processes. multivariate Wiener process. Periodic inspection is performed degradation-based maintenance is only one degradation The dependent degradation processes are modeled as a process imposes impact ontothe system. However, in reality, degradation-based maintenance is that that multiple only one functions, degradation as a product is intended perform it multivariate The dependent degradation processes are modeled as process. Periodic inspection is performed process imposes impact on the system. However, in reality, Thereveal dependent degradation processes are When modeled as aa theWiener existence of hidden failures. a hidden process imposes impact on system. However, in reality, multivariate Wiener process. Periodic inspection is performed as a product ismultiple intended tothe perform multiple functions, it to process imposes impact on the system. However, in reality, usually suffers degradation processes. multivariate process. Periodic inspection is to reveal theWiener existence of hidden failures. a hidden as aa product is intended to multiple Wiener process. Periodic inspection is performed performed failure is discovered at inspection, the systemWhen is replaced and as product ismultiple intended to perform perform multiple functions, functions, it it multivariate to reveal the existence of hidden failures. When aa hidden usually suffersis degradation processes. as a product intended to perform multiple functions, it to reveal the existence of hidden failures. When hidden failure is discovered at inspection, the system is replaced and usually suffers multiple degradation processes. reveal the existence of hidden failures. When a hidden afailure renewal process takes place. Optimal inspection interval is In addition, degradation processesprocesses. are usually mutually to usually suffers multiple degradation is discovered at inspection, the system is replaced and usually suffersdegradation multiple degradation processes. failure is discovered at inspection, the system is replaced and a renewal process takes place. Optimal inspection interval is In addition, processes are usually mutually failure is discovered at inspection, the system is replaced and obtained by minimizing the long-run cost rate. correlated (Wang and Pham, 2012). For example, in a rail a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually obtained by minimizing the long-run cost rate. correlated (Wang and Pham, 2012). For example, in a rail a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually track, there(Wang may exist more than one For cracks, and each crack obtained by minimizing the long-run cost rate. correlated and Pham, 2012). example, in a rail The remainder of the paper organized follows. Section 2 obtained by minimizing minimizing the is long-run costasrate. rate. correlated (Wang and more Pham, 2012). For example, in if acrack rail track,suffer there(Wang may exist than one cracks, and each obtained by the long-run cost correlated and Pham, 2012). For example, in a rail may a degradation process. The rail track fails the The of the paper is organized as follows. Section 2 track, there exist than cracks, and crack givesremainder the description of degradation process and reliability track,suffer there amay may exist more moreprocess. than one one cracks, and each each crack remainder of the paper is organized as follows. Section 2 may degradation Thecracks, rail track fails if the The track, there may exist more than one and each crack The remainder of the paper is organized as follows. Section gives the description of degradation process and reliability may suffer a degradation process. The rail track fails if the The remainder of the paper is organized as follows. Section 2 2 may the description of degradation process and reliability may suffer suffer aa degradation degradation process. process. The The rail rail track track fails fails if if the the gives gives gives the the description description of of degradation degradation process process and and reliability reliability
Copyright © 2016 IFAC 1377 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 1377Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 1377Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC IFAC 1377 Copyright © 2016 1377 10.1016/j.ifacol.2016.07.758
IFAC MIM 2016 1378 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
model. The cost model is developed in Section 3 and the algorithm to obtain the optimal inspection interval is presented in Section 4. In Section 5, a numerical example is given to illustrate the degradation process and inspection policy. Finally, conclusions are drawn in Section 6.
Denote K 2 as the number of inspections after a failure occurs. Denote N f (t ) as the number of failures by time t.
N f (t ) takes values in 0,1, 2,...m with the probabilities as
X k (t ) H k , k U j P( N f (t ) j ) P Uj X ( t ) H , l U l k j
2. DEGRADATION DESCRIPTION In this study, we consider a one-unit system. It is assumed that the system comprises m degradation processes, which can be represented as a multivariate Wiener degradation process (Barker and Newby, 2009), X(t ) λt V(t ) , where X(t ) is the vector of degradation level by time t, λ denotes the vector of drift coefficients and V (t ) is a vector of random parameters, describing the uncertainty of the degradation processes. Specifically, V (t ) follows a multivariate normal distribution, N (0, t) , where is the variance-covariance matrix. System reliability is defined as the probability that all the degradation levels stay within pre-fixed thresholds, i.e., R(t ) P X1 H1 , ..., X m H m . Suppose the variance of the degradation process is much smaller than the drift coefficient, then the probability that the degradation level X i (t ) is negative can be neglected. Reliability of the system is expressed as
(4)
where Uj is the failure set containing j failures. (4) measures all the scenarios that j failures have occurred. If j failures occur, the probability that at least one failure can be discovered at inspection is 1 j , where is the probability that a failure cannot be detected at an inspection. Given that the system has already failed, the probability that failure can be detected at an inspection is given as m
Pd (t | t T f )
PN
f
j 1
(t ) j (1 j ) 1 R(t )
(5)
For notational simplicity, denote Pd (t | t T f ) as Pd (t ) . Given that K1 k1 , we have i 1
P K 2 =i | K1 k1 Pd (k1T iT ) 1 Pd (k1T jT )
(6)
j 0
The unconditioned probability of K 2 can be obtained as
m
1 Hm 1 2 2 R(t ) ... t 2 T 1 exp x(t ) λt 1 x(t ) λt dx1 ...dxm 2t H1
P K 2 =i P K 2 =i | K1 k1 P K1 k1
(1)
The expectation of K 2 is expressed as
3. COST MODEL The cost model is formulated in terms of inspection interval, where the long run cost rate is achieved by use of the renewal cycle theorem. A renewal cycle is defined as the interval between two consecutive replacements or the interval between installment and the first replacement (Peng et al, 2010; Liu and Huang, 2010). In the present model, the cost items include inspection cost, downtime cost and replacement cost. Inspection cost occurs each time an inspection is carried out and downtime cost is incurred when the system is operating with the existence of failure (Berrade et al, 2012). Let K1 be the number of inspections before a failure occurs. We have P( K1 i) P Tf [iT ,(i 1)T ] R iT R (i 1)T
interval. The expected value of K1 can be obtained as
i 0
i 1
E[ K1 ] iP( K1 i ) R(iT )
E[ K 2 ] iP K 2 =i i 1
i R k1T R (k1 1)T k1 0 i 1 i 1 Pd (k1T iT ) 1 Pd (k1T jT ) j 0
(8)
As failure can only be detected at inspection and the system is replaced when a failure is discovered, the length of a renewal cycle is equal to the time interval from beginning to the last inspection. Therefore, we can obtain the expected length of a renewal as E[S ] E[ K1 ] E[ K 2 ]T
(9)
(2)
where T f is the time to failure and T is the inspection
(7)
k1 0
Denote Td as the system downtime, time interval between failure occurrence and system replacement. Expectation of Td can be expressed as E[Td ] E[S ] E[Tf ] . The expected cost within a renewal cycle is given as
E[C ] CR CI E[ K1 K2 ] Cd E[Td ]
(3)
(10)
where CR is the replacement cost, CI is the inspection cost, and Cd is the downtime cost per unit time.
1378
IFAC MIM 2016 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
Combining (9) and (10), we can have the expected long run cost rate as C (t ) E[C ] CR(T ) lim t t E[ S ] CR Cd E[T f ] C Cd I T E[ K1 K 2 ] T
CI CR Cd tdR (t ) / (11) 0 T i R k1T R (k1 1)T k1 0 T R(iT ) i 1 i 1 i 1 Pd (k1T iT ) 1 Pd (k1T jT ) j 0 Cd
4. OPTIMIZATION ALGORITHM Our goal is to determine the optimal T that minimizes the objective function CR(T), i.e.,
T arg min CR(T ) T
(12)
Analytical solution of CR(T) is hard to obtain due to the fact that E[ K1 K2 ] depends on both the degradation process and inspection interval T. Hence, we resort to Monte Carlo Simulation to calculate CR(T) (Huynh et al., 2012a). When the number of simulation histories Nl is large enough, (11) can be rewritten as Nl
Nl
n 1
n 1
CR(T ) C ( n ) / S ( n )
Combining (14) and (15), we have
E[T f ]
mT T E[ K1 K 2 ] T E[T f ] m 1 1
(16)
The proof can be concluded by substituting (16) into (11). Proposition 1 is useful in engineering, as engineers are more interested in the bounds rather than the exact value. Although the analytical solution of CR(T) is difficult to get, the upper and lower bounds of CR(T) can be obtained as an analytical solution. In situations where computational efficiency is the main concern, the bounds can be adopted instead of the exact value. Define CR Cd E[T f ] C f1 (T ) Cd I T mT E[T f ] 1 m and C CR Cd E[T f ] f 2 (T ) Cd I T T E[T f ] 1 More properties of CR(T) can be concluded as follows: Corollary. CR(T) converges to Cd for T . Proof. Based on the definition of f1 (T ) and f 2 (T ) , it is straightforward
to
see
that
lim f1 (T ) Cd
T
and
lim f 2 (T ) Cd . According to Proposition 1, we can have
T
(13)
where C(n) and S(n) are respectively the cost and length of a renewal cycle for the nth simulation history. In what follows, we design a search algorithm to determine the optimal T and CR(T).
lim CR(T ) Cd .
T
Intuitively, the corollary can be interpreted as follows: if T , then no inspection is performed for the system. As a result, the system is left failed and the long run cost rate approaches Cd .
Assume CR Cd E[Tf ] and CI Cd E[Tf ] . This assumption is reasonable in reality due to the fact that unexpected failures incur higher cost than replacement and inspection (Huynh et al, 2012a; Van Oosterom et al, 2014). Several properties of the objective function are derived for optimization algorithm. The following proposition gives the upper bound and lower bound of the objective function. Proposition 1. The long run cost rate CR(T) is limited in the interval as CR Cd E[T f ] C C CR Cd E[T f ] CR(T ) Cd I Cd I m T T E[T ] T T E[T f ] f m 1 1 Proof. K1 can be rewritten as
Tf Tf 1 K1 (14) T T T As K2 is inversely related to the number of failures at inspection and the number of failures is limited as 1 N f m , we can obtain Tf
1 1 E[ K 2 ] 1 1 m
1379
(15)
Lemma.
f1 (T ) and
f 2 (T ) are unimodel functions for
T 0, . Proof. Rewrite f 2 (T ) as a b T cT d 1 , d E[T f ] . We can where a CI , b Cd E[T f ] CR , c 1 f 2 (T ) Cd
obtain the derivative of f 2 (T ) as f 2' (T )
a bc T 2 cT d 2
Let f 2' (T ) 0 , we have
bc ac T 2
2
2acdT ad 2 0
According to the assumption that b ac , we can conclude that only one root exists over the range of 0, . Denote the root as Tu . For T 0, Tu , f 2 (T ) is monotonously
1379
IFAC MIM 2016 1380 June 28-30, 2016. Troyes, France
decreasing;
then
for
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
T Tu , ,
f 2 (T )
increases
monotonously. Combined with lim f 2 (T ) , it can be T 0
concluded that f 2 (T ) is a unimodel function for T 0, . Similarly, we can conclude f1 (T ) is a unimodel function for
T 0, . The unimodality of f1 (T ) and f 2 (T ) for T 0, is important as the property helps to determine the range of the optimum inspection interval T. Let f1 denote the minimum 2
of f1 (T ) and f be the minimum of f 2 (T ) . We can have the following proposition to determine the bounds of the optimum T. Proposition 2. The optimal inspection interval T lies in the range as T Ta , Tb , where Ta and Tb are the roots of the equation f1 (T ) f 2 ,
Ta , Tb arg f1 (T ) f 2 . T
Proof. First, we need to prove that such Ta and Tb exist. Proposition 1 shows that f1 (T ) f 2 (T ), T (0, ) . Hence, we have f1 f 2 . Also, lim f1 (T ) f 2 and T 0
lim f1 (T ) Cd f 2 . Along with the unimodality of f1 (T ) ,
T
5. AN ILLUSTRATIVE EXAMPLE A system with two fatigue crack positions is considered as an example to illustrate the proposed inspection policy. The original measurements of fatigue crack sizes are collected from Pan and Balakrishnan (2011). The data has been used for degradation modeling and reliability estimation in Pan et al (2013). In this paper, we use the same data to estimate the parameters of degradation process and further make inspection decisions. Pan et al (2013) pointed out that the aforementioned degradation process is not linearly increasing in term of time unit. A bivariate Wiener process with an exponential time
transform t is used to model the deterioration process (2013). The parameter is set to be 1.3 (Pan et al, 2013). We fit the data for the degradation model and estimate the other parameters by maximum likelihood estimation (MLE). The parameters are estimated as 1 12.2187 , 2 7.5967 , [0.0505,0.0147;0.0147,0.0247] . The parameters are used as true values in this study. The system fails if any of the two cracks exceeds 1.6 in. in size. The initial degradation levels of the two cracks are 0.9 respectively. To neutralize the effect of initial degradation, the failure threshold is set as H1 H 2 0.7 . By use of the estimates of model parameters, system reliability can be calculated based on (1). Fig. 1 shows the variation of system reliability in terms of .
we can conclude that there exist Ta and Tb that satisfy
Ta , Tb arg f1 (T ) f 2 . Let f 2 (T ) reaches the minimum at T
T Tu . The result can be proved by contradiction. Suppose T were larger than Tb , T Tb , it could be shown that CR(T ) f1 (T ) f1 (Tb ) f 2 . However, according to Proposition 1, we have CR(T ) CR(Tu ) f 2 (Tu ) f 2 , which is a contradiction. Therefore, it can be concluded that T Tb . Similarly, we can conclude that T Ta . Proposition 2 gives the lower and upper bounds of the search algorithm. Base on Proposition 2, we can design the optimization algorithm as follows. Algorithm: Optimization procedure Step 1: start from T=Ta. Step 2: Monte Carlo simulation to calculate the long-run cost rate. 2.1: initialization: set C = 0 and S = 0. 2.2: generate a history of degradation levels X(t). 2.3: compute the cost and length of a renewal cycle, C(n) a nd S(n). 2.4: compute C = C + C(n), S = S + S(n), and CR = C/S. If CR converges, go to step 3; otherwise, repeat steps 2.1-2.4. Step 3: if T
Fig. 1 Plot of system reliability when varies The cost parameters are set as CI 2 , CR 50 and Cd 5000 . The probability that a failure cannot be detected by an inspection is related to the inspection technique. In this study, is set to be 0.5. According to Proposition 2, we obtain the lower and upper bound of the optimal inspection interval as Ta 0.0021 and Tb 0.0527 . By searching within the range Ta , Tb , the minimum expected cost rate is obtained as CR 1512 , at the optimum inspection interval T 4.4 103 . Fig. 2 shows the variation of expected cost rate at different inspection intervals.
1380
IFAC MIM 2016 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
1381
appropriate inspection technique and frequency can be determined. Future research can be conducted from the following two aspects. First, in this paper, the detection probability is assumed to be constant. However, in reliability, the detection probability is not only related to inspection techniques, but also to the degradation level. It would be more reasonable to model the detection probability as a function of the degradation level. Second, other inspection policies (e.g., non-periodic inspection) are of interest to investigate. REFERENCES Fig. 2 Plot of cost rate for different inspection intervals
Sensitivity analysis is performed on in order to study the effect of different inspection techniques on the expected cost rate. Fig. 3 plots the variation of CR(T ) and T for different values of . As shown in Fig. 3, the minimum cost rate increases from 1285 to 2186 when varies from 0.1 to 0.9. The result implies that inspection accuracy has a significant impact on the cost rate. Engineers or managers are suggested to pay more efforts to improve the inspection quality so as to reduce the cost generated by hidden failures.
Fig. 3 Sensitivity analysis of CR(T ) and T on 6. CONCLUSIONS In this study, an inspection policy is proposed for systems subject to multiple degradation processes. The inspection is assumed to be imperfect, that a failure can only be discovered at inspection with certain probability. System reliability is analyzed, followed by a cost model as the objective function. Optimal inspection interval is determined by minimizing the long run cost rate. The result suggests that engineers or managers should invest on the improvement of inspection techniques in order to reduce the operation cost. This study can be applied in many practical problems, for example, to detect the crack size in railway tracks or bridges, engineers have many options such as ultrasonic testing, visual inspection and magnetic particle inspection. It is desirable to determine the most proper inspection technique and the associated inspection frequency. By balancing the cost and benefits with various testing options, the most
Barker, C. T., and Newby, M. J. (2009). Optimal nonperiodic inspection for a multivariate degradation model. Reliability Engineering and System Safety, 94(1), 33-43. Berrade, M. D. (2012). A two-phase inspection policy with imperfect testing. Applied Mathematical Modelling, 36(1), 108-114. Berrade, M. D., Cavalcante, C. A., and Scarf, P. A. (2012). Maintenance scheduling of a protection system subject to imperfect inspection and replacement. European Journal of Operational Research, 218(3), 716-725. Berrade, M. D., Scarf, P. A., Cavalcante, C. A. V., and Dwight, R. A. (2013). Imperfect inspection and replacement of a system with a defective state: A cost and reliability analysis. Reliability Engineering and System Safety, 120, 80-87. Huynh, K. T., Barros, A., and Berenguer, C. (2012a). Maintenance decision-making for systems operating under indirect condition monitoring: value of online information and impact of measurement uncertainty. IEEE Transactions on Reliability, 61(2), 410-425. Huynh, K. T., Castro, I. T., Barros, A., and Berenguer, C. (2012b). Modeling age-based maintenance strategies with minimal repairs for systems subject to competing failure modes due to degradation and shocks. European Journal of Operational Research, 218(1), 140-151. Li, W., and Pham, H. (2005). An inspection-maintenance model for systems with multiple competing processes. IEEE Transactions on Reliability, 54(2), 318-327. Liu, B., Xu, Z., Xie, M., and Kuo, W. (2014). A Value-based Preventive Maintenance Policy for Multi-component System with Continuously Degrading Components. Reliability Engineering and System Safety, 132, 83-89. Liu, X., Li, J., Al-Khalifa, K. N., Hamouda, A. S., Coit, D. W., and Elsayed, E. A. (2013). Condition-based maintenance for continuously monitored degrading systems with multiple failure modes. IIE Transactions, 45(4), 422-435. Liu, Y., and Huang, H. Z. (2010). Optimal replacement policy for multi-state system under imperfect maintenance. IEEE Transactions on Reliability, 59(3), 483-495. Pan, Z., and Balakrishnan, N. (2011). Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes. Reliability Engineering and System Safety, 96(8), 949-957. Pan, Z., Balakrishnan, N., Sun, Q., and Zhou, J. (2013). Bivariate degradation analysis of products based on
1381
IFAC MIM 2016 1382 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
Wiener processes and copulas. Journal of Statistical Computation and Simulation, 83(7), 1316-1329. Peng, H., Feng, Q., and Coit, D. W. (2010). Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes. IIE transactions, 43(1), 12-22. Van Oosterom, C. D., Elwany, A. H., Çelebi, D., and van Houtum, G. J. (2014). Optimal policies for a delay time model with postponed replacement. European Journal of Operational Research, 232(1), 186-197. Wang, X., Balakrishnan, N., and Guo, B. (2014). Residual life estimation based on a generalized Wiener degradation process. Reliability Engineering and System Safety, 124, 13-23. Wang, Y., and Pham, H. (2012). Modeling the dependent competing risks with multiple degradation processes and random shock using time-varying copulas. IEEE Transactions on Reliability, 61(1), 13-22. Ye, Z. S., and Xie, M. (2015). Stochastic modelling and analysis of degradation for highly reliable products. Applied Stochastic Models in Business and Industry, 31(1), 16-32. Ye, Z. S., Xie, M., Tang, L. C., and Shen, Y. (2012). Degradation-based burn-in planning under competing risks. Technometrics, 54(2), 159-168.
1382
ScienceDirect
IFAC-PapersOnLine 49-12 (2016) 1377–1382 IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE CORRELATED DEGRADATION PROCESSES IMPERFECT INSPECTION POLICY FOR SYSTEMS WITH IMPERFECTCORRELATED INSPECTION POLICY FOR SYSTEMS WITH MULTIPLE MULTIPLE DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES CORRELATED DEGRADATION PROCESSES Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo*
Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo* Bin Liu*, Xiujie Zhao*, Ruey-Huei Yeh**, Way Kuo* Bin Liu*, Zhao*, Ruey-Huei Yeh**, Bin Engineering Liu*, Xiujie Xiujieand Zhao*, Ruey-Huei Yeh**, Way Way Kuo* * Department of Systems Engineering Management, CityKuo* University of Hong Kong, * Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) * Department of Systems Engineering and Engineering Management, City University of Hong Kong, *** Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) * Department of Systems Engineering and Engineering Management, City University of Hong Kong, Department of(e-mail: Industrial Management, National Taiwan University of Science [email protected]) Technology, Kowloon, Hong Kong [email protected]; [email protected]; Kowloon, Hong Kong [email protected]; [email protected]; ** Department of(e-mail: Industrial Management, National Taiwan University of Science [email protected]) Technology, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]) Taipei, Taiwan (email: [email protected])} ** Management, National Taiwan ** Department Department of of Industrial Industrial Management, National Taiwan University University of of Science Science and and Technology, Technology, Taipei, Taiwan (email: [email protected])} ** Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan (email: [email protected])} Taipei, Taipei, Taiwan Taiwan (email: (email: [email protected])} [email protected])} Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated Abstract: This paper develops an imperfect inspection policy for systems subject toWiener multiple correlated degradation processes. The degradation processes are characterized by a multivariate process, and Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated degradation processes. The degradation processes are characterized by a multivariate Wiener process, and Abstract: This paper develops an imperfect inspection policy for systems subject to multiple correlated the dependency among the processes is described by a covariance matrix. A failure occurs when any of degradation processes. The degradation processes are characterized by aa multivariate Wiener process, and degradation processes. The are characterized by multivariate Wiener process, the dependency among the degradation processes is processes described byAs a covariance matrix. A failureare occurs when anyand of degradation processes. The degradation processes are characterized by a multivariate Wiener process, and the degradation level exceeds a specific threshold. degradation-based failures usually dormant, the dependency among the processes is described by aa covariance matrix. A failure occurs when any of the dependency among processes is by matrix. A failure occurs when any degradation level exceeds athe specific threshold. As degradation-based failures are usually dormant, the dependency among the processes is described described byAs a covariance covariance matrix.The A failure occurs when any of of inspection is required tothe detect failures and prevent potential losses. inspection is imperfect, in the degradation level exceeds a specific threshold. degradation-based failures are usually dormant, the degradation level exceeds aathe specific threshold. As degradation-based failures are usually dormant, inspection isthat required to detect failures and prevent potential Optimal losses. The inspection is imperfect, in the degradation level exceeds specific threshold. As degradation-based failures are usually dormant, such a way a failure may not be discovered at an inspection. inspection interval is obtained inspection is required to detect the failures and potential losses. The inspection is in inspection isthat required to run detect therate. failures and prevent prevent potential Optimal losses. The inspection is imperfect, imperfect, in such a wayis a failure maycost not befailures discovered at an inspection. inspection interval is obtained inspection required to detect the and prevent potential losses. The inspection is imperfect, in by minimizing the long In addition, theinspection. properties of the optimal inspection interval are such a way that a failure may not be discovered at an Optimal inspection interval is obtained such aa way that aa failure may not be discovered at an Optimal inspection interval is obtained by minimizing the long run cost rate. Inofaddition, theinspection. properties of the optimal inspection interval are such way that failure may not be discovered at an inspection. Optimal inspection interval is obtained studied and the upper and lower bounds the optimum value are achieved. An example of fatigue crack by minimizing the long run cost rate. In the properties of optimal inspection interval are by minimizing the long run costbounds rate. Inofaddition, addition, the properties of the the optimal inspection interval are studied and the upper andillustrate lower the optimum value are achieved. An example of fatigue crack by minimizing run cost rate. In addition, properties of the inspection interval are development is the usedlong to and evaluate the the inspection policy. Theoptimal results imply of that inspection studied and the upper and lower bounds of the optimum value are achieved. An example fatigue crack studied and the and lower bounds of the optimum value are achieved. An example of fatigue crack development is upper used significantly to illustrate and evaluate the operating inspection policy. The results imply that inspection studied and the upper and lower bounds of the optimum value are achieved. An example of fatigue crack accuracy contributes to reducing the cost and more efforts should be made to development is used to illustrate and evaluate the inspection policy. The results imply that inspection development is used to illustrate and evaluate the inspection policy. imply that accuracy contributes significantly to reducing operating cost and The moreresults efforts should beinspection made to development is used to illustrate and evaluate the inspection policy. The results imply that inspection improve the inspection quality. accuracy contributes significantly to reducing the operating cost and more efforts should be made to accuracy contributes to the cost and efforts should be improve the inspectionsignificantly quality. accuracy contributes significantly to reducing reducing the operating operating cost hidden and more more efforts shouldanalysis be made made to to improve the inspection quality. Keywords: Imperfect inspection; correlated degradation processes; failure; reliability © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. improve the inspection quality. improve theImperfect inspectioninspection; quality. correlated degradation processes; hidden failure; reliability analysis Keywords: Keywords: processes; Keywords: Imperfect Imperfect inspection; inspection; correlated correlated degradation degradation processes; hidden hidden failure; failure; reliability reliability analysis analysis Keywords: Imperfect inspection; correlated degradation processes; hidden failure; reliability analysis size of any crack exceeds a specific threshold. The cracks 1. INTRODUCTION size of correlated any crack inexceeds a specific cracks may be such a way that allthreshold. the cracksThe are subject size of any crack exceeds aa specific threshold. The cracks 1. INTRODUCTION size of any crack specific The cracks may be correlated inexceeds such awheel way that allthreshold. theand cracks are subject 1. INTRODUCTION size of any crack exceeds a specific threshold. The cracks to common shocks, e.g., fraction environmental Nowadays, maintenance has gained increasing attention as 1. INTRODUCTION may be correlated in such a way that all the cracks are subject 1. INTRODUCTION may be correlated in such a way that all the cracks are subject to common shocks, e.g., wheel fraction and environmental Nowadays, maintenance hasimprove gained increasing attention as may be correlated in such a way that all the cracks are subject corrosion. Maintenance policy on multiple correlated maintenance operations can production quality and to common shocks, e.g., wheel fraction and environmental Nowadays, maintenance has gained increasing attention as to common shocks, wheel fraction and Nowadays, maintenance has gained as corrosion. Maintenance policy onresearch. multiple correlated maintenance operations can improve production quality and to common processes shocks, e.g., e.g., wheel fraction and environmental environmental requires further Nowadays, maintenance has gained increasing increasing attention as degradation reduce losses caused by unexpected failures. attention Traditional corrosion. Maintenance policy on multiple correlated maintenance operations can improve production quality and corrosion. Maintenance policy on multiple maintenance operations can improve production quality and degradation processes requires further research. reduce losses caused by unexpected failures. Traditional corrosion. processes Maintenance policy onresearch. multiple correlated correlated operations can improvebased production quality and degradation maintenance actions areby performed on failure models, requires further reduce losses caused unexpected failures. Traditional Failure caused by degradation is referred to as “hidden degradation processes requires further research. reduce losses caused by unexpected failures. Traditional maintenance actions are performed based on In failure models, degradation processes requires further research. reduce losses caused by unexpected failures. Traditional which are constructed by use of failure data. recent years, Failure or caused by degradation isal, referred to asis “hidden maintenance actions are performed based on failure models, failure” “soft failure” (Peng et 2010), which usually maintenance actions are based on failure models, caused by degradation is referred to as “hidden which aredue constructed byperformed use of failure data. recent years, Failure maintenance actions are performed based on In failure models, however, to improved quality and reliability of products, Failure caused by degradation to as failure” or “soft failure” (Peng et is al,bereferred 2010), which is “hidden usually which are constructed by use of failure data. In recent years, Failure caused by degradation is referred to as “hidden dormant to engineers and can only revealed by inspection which constructed by of data. years, or “soft failure” (Peng et al, 2010), which is usually however, due to improved quality and reliability of which are constructed by use use of failure failure data. In InIt recent recent years, failure” failure are data are becoming harder to obtain. is products, therefore failure” or “soft failure” (Peng et al, 2010), which is usually dormant to engineers and can only be revealed by inspection however, due to improved quality and reliability of products, failure” or “soft failure” (Peng et al,bearticles 2010), which is that usually or testing. Majority ofand the existing assume the however, due to improved improved quality and reliability of products, dormant to engineers can only revealed by inspection failure data are becoming harder and to obtain. It of is on therefore however, due to quality reliability products, difficult to implement maintenance activities based failure dormant to engineers and can only be revealed by inspection or testing. Majority of the existing articles assume that the failure data are becoming harder to obtain. It is therefore dormant to engineers and can only be revealed by inspection inspection is perfect. However, in reality, inspection is failure data data are Xie, becoming harder activities to obtain. obtain.based It is is on therefore testing. Majority of the existing articles assume that the difficult to implement maintenance failure or failure are becoming harder to It therefore models (Ye and 2015). or testing. Majority of the existing articles assume that the inspection is perfect. However, in reality, inspection is difficult to implement maintenance activities based on failure or testing. Majority of the existing articles assume that the usually less than perfect; the hidden failure cannot always be difficult to implement maintenance activities based on failure is perfect. However, in reality, inspection is models and Xie, 2015). difficult(Ye to implement maintenance activities based on failure inspection inspection is perfect. However, in reality, inspection is usually less than perfect; the hidden failure cannot always be models (Ye and Xie, 2015). inspection is perfect. However, in reality, inspection is spotted less by an inspection (Berrade etfailure al, 2013). During one As an alternative, degradation models provide opportunities usually models (Ye and Xie, 2015). than perfect; the hidden cannot always be models (Ye and Xie, 2015). usually less than perfect; the hidden failure cannot always be spotted by an inspection (Berrade et al, 2013). During one As an alternative, degradation models provide opportunities usually less than perfect; the hidden failure cannot always be inspection, the hidden failure can only be detected with some for an reliability analysis and maintenance decisionopportunities making for spotted by an inspection (Berrade et al, 2013). During one As alternative, degradation models provide spotted probability. by the an inspection inspection (Berrade et al, al, 2013). During During one As an an alternative, degradation modelsDegradation provide opportunities inspection, hidden failure can only be detected with some for reliability analysis and decisionopportunities making for spotted by an (Berrade et 2013). one certain As alternative, degradation models provide highly reliable products andmaintenance systems. models are inspection, the hidden failure can only be detected with some for reliability analysis and maintenance decision making for inspection, the hidden failure can only be detected with some for reliability analysis and decision making for certain probability. highly reliable products andmaintenance systems. Degradation models are inspection, the hidden failure can only be detected with some for reliability analysis and maintenance decision making for constructed by use of degradation measurements, which can certain probability. highly reliable products and systems. Degradation models are No existing maintenance strategy has covered the joint effect certain probability. highly reliable products and Degradation are constructed by use operation of degradation measurements, which certain probability. highly reliable products and systems. systems. Degradation models are be recorded during of a product or systemmodels (Ye etcan al, No existing maintenance processes strategy has the joint effect constructed by use of degradation measurements, which can of multiple degradation andcovered imperfect inspection. constructed by use of degradation measurements, which can No existing maintenance strategy has covered the effect be recorded during operation of a product or system (Ye et al, constructed by use of degradation measurements, which can 2012). Another advantage ofofdegradation model lies in its of Nomultiple existing maintenance strategy has covered the joint joint effect degradation processes andcovered imperfect inspection. be recorded during operation a product or system (Ye et al, No existing maintenance strategy has the joint effect Our research aims to fulfill the gap and contribute to the be recorded during operation of a product or system (Ye et al, of multiple degradation processes and imperfect inspection. 2012). Another advantage of degradation model lies in its be recorded operation ofdegradation a product system (Yein et al, ability to during characterize physical failureormodel mechanism. In of multiple degradation processes and imperfect inspection. Our researchof aims to fulfill the gap and contribute to the 2012). Another advantage of lies its of multiple degradation processes and imperfect inspection. knowledge maintenance concerning multiple dependent 2012). Another advantage of degradation model lies research aims to fulfill the gap and contribute to the ability todegradation-based characterize failure mechanism. In 2012). Another advantagephysical of model degradation model lies in in its its addition, provides flexibility to Our Our research aims to the contribute to knowledge maintenance dependent ability to characterize physical failure mechanism. In Our researchof aims to fulfill fulfill the gap gap aand andmultiple contribute to the the degradation processes by concerning adding realistic factor of ability to characterize physical failure mechanism. In knowledge of maintenance concerning multiple dependent addition, degradation-based model provides flexibility to ability todegradation-based characterize failure mechanism. In investigate the impact ofphysical environmental variation on the knowledge of maintenance degradation processes by concerning adding a multiple realistic dependent factor of addition, model provides flexibility to knowledge of maintenance concerning multiple dependent imperfect inspection. addition, degradation-based model provides provides flexibility to degradation processes by adding a realistic factor of investigate the impact of environmental variation on the addition, degradation-based model flexibility to deteriorating process. degradation processes imperfect inspection. investigate the impact of environmental variation on the degradation processes by by adding adding aa realistic realistic factor factor of of investigate impact of environmental variation on the imperfect inspection. deterioratingthe process. investigate the impact of environmental variation on the In this study, we develop an imperfect inspection policy for imperfect inspection. deteriorating process. imperfect inspection. An implicit assumption of most existing works on deteriorating process. In this study, imperfectdegradation inspection processes. policy for deteriorating subjectwe to develop multiplean correlated In this study, we develop an imperfect inspection policy for An implicit process. assumption of ismost existing works on system degradation-based maintenance that only one degradation In this study, develop imperfect inspection policy system subjectwe todegradation multiplean correlated degradation processes. An implicit assumption of most existing works on In this study, we develop an imperfect inspection policyasfor fora The dependent processes are modeled An implicit assumption of most existing works on system subject to multiple correlated degradation processes. degradation-based maintenance is that only one degradation An implicit of system. most existing works on system process imposesassumption impact on the However, in reality, subject to multiple correlated degradation processes. The dependent degradation processes are modeled as a degradation-based maintenance is that only one degradation system subject to multiple correlated degradation processes. multivariate Wiener process. Periodic inspection is performed degradation-based maintenance is only one degradation The dependent degradation processes are modeled as a process imposes impact ontothe system. However, in reality, degradation-based maintenance is that that multiple only one functions, degradation as a product is intended perform it multivariate The dependent degradation processes are modeled as process. Periodic inspection is performed process imposes impact on the system. However, in reality, Thereveal dependent degradation processes are When modeled as aa theWiener existence of hidden failures. a hidden process imposes impact on system. However, in reality, multivariate Wiener process. Periodic inspection is performed as a product ismultiple intended tothe perform multiple functions, it to process imposes impact on the system. However, in reality, usually suffers degradation processes. multivariate process. Periodic inspection is to reveal theWiener existence of hidden failures. a hidden as aa product is intended to multiple Wiener process. Periodic inspection is performed performed failure is discovered at inspection, the systemWhen is replaced and as product ismultiple intended to perform perform multiple functions, functions, it it multivariate to reveal the existence of hidden failures. When aa hidden usually suffersis degradation processes. as a product intended to perform multiple functions, it to reveal the existence of hidden failures. When hidden failure is discovered at inspection, the system is replaced and usually suffers multiple degradation processes. reveal the existence of hidden failures. When a hidden afailure renewal process takes place. Optimal inspection interval is In addition, degradation processesprocesses. are usually mutually to usually suffers multiple degradation is discovered at inspection, the system is replaced and usually suffersdegradation multiple degradation processes. failure is discovered at inspection, the system is replaced and a renewal process takes place. Optimal inspection interval is In addition, processes are usually mutually failure is discovered at inspection, the system is replaced and obtained by minimizing the long-run cost rate. correlated (Wang and Pham, 2012). For example, in a rail a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually obtained by minimizing the long-run cost rate. correlated (Wang and Pham, 2012). For example, in a rail a renewal process takes place. Optimal inspection interval is In addition, degradation processes are usually mutually track, there(Wang may exist more than one For cracks, and each crack obtained by minimizing the long-run cost rate. correlated and Pham, 2012). example, in a rail The remainder of the paper organized follows. Section 2 obtained by minimizing minimizing the is long-run costasrate. rate. correlated (Wang and more Pham, 2012). For example, in if acrack rail track,suffer there(Wang may exist than one cracks, and each obtained by the long-run cost correlated and Pham, 2012). For example, in a rail may a degradation process. The rail track fails the The of the paper is organized as follows. Section 2 track, there exist than cracks, and crack givesremainder the description of degradation process and reliability track,suffer there amay may exist more moreprocess. than one one cracks, and each each crack remainder of the paper is organized as follows. Section 2 may degradation Thecracks, rail track fails if the The track, there may exist more than one and each crack The remainder of the paper is organized as follows. Section gives the description of degradation process and reliability may suffer a degradation process. The rail track fails if the The remainder of the paper is organized as follows. Section 2 2 may the description of degradation process and reliability may suffer suffer aa degradation degradation process. process. The The rail rail track track fails fails if if the the gives gives gives the the description description of of degradation degradation process process and and reliability reliability
Copyright © 2016 IFAC 1377 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 1377Hosting by Elsevier Ltd. All rights reserved. Copyright 2016 responsibility IFAC 1377Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC IFAC 1377 Copyright © 2016 1377 10.1016/j.ifacol.2016.07.758
IFAC MIM 2016 1378 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
model. The cost model is developed in Section 3 and the algorithm to obtain the optimal inspection interval is presented in Section 4. In Section 5, a numerical example is given to illustrate the degradation process and inspection policy. Finally, conclusions are drawn in Section 6.
Denote K 2 as the number of inspections after a failure occurs. Denote N f (t ) as the number of failures by time t.
N f (t ) takes values in 0,1, 2,...m with the probabilities as
X k (t ) H k , k U j P( N f (t ) j ) P Uj X ( t ) H , l U l k j
2. DEGRADATION DESCRIPTION In this study, we consider a one-unit system. It is assumed that the system comprises m degradation processes, which can be represented as a multivariate Wiener degradation process (Barker and Newby, 2009), X(t ) λt V(t ) , where X(t ) is the vector of degradation level by time t, λ denotes the vector of drift coefficients and V (t ) is a vector of random parameters, describing the uncertainty of the degradation processes. Specifically, V (t ) follows a multivariate normal distribution, N (0, t) , where is the variance-covariance matrix. System reliability is defined as the probability that all the degradation levels stay within pre-fixed thresholds, i.e., R(t ) P X1 H1 , ..., X m H m . Suppose the variance of the degradation process is much smaller than the drift coefficient, then the probability that the degradation level X i (t ) is negative can be neglected. Reliability of the system is expressed as
(4)
where Uj is the failure set containing j failures. (4) measures all the scenarios that j failures have occurred. If j failures occur, the probability that at least one failure can be discovered at inspection is 1 j , where is the probability that a failure cannot be detected at an inspection. Given that the system has already failed, the probability that failure can be detected at an inspection is given as m
Pd (t | t T f )
PN
f
j 1
(t ) j (1 j ) 1 R(t )
(5)
For notational simplicity, denote Pd (t | t T f ) as Pd (t ) . Given that K1 k1 , we have i 1
P K 2 =i | K1 k1 Pd (k1T iT ) 1 Pd (k1T jT )
(6)
j 0
The unconditioned probability of K 2 can be obtained as
m
1 Hm 1 2 2 R(t ) ... t 2 T 1 exp x(t ) λt 1 x(t ) λt dx1 ...dxm 2t H1
P K 2 =i P K 2 =i | K1 k1 P K1 k1
(1)
The expectation of K 2 is expressed as
3. COST MODEL The cost model is formulated in terms of inspection interval, where the long run cost rate is achieved by use of the renewal cycle theorem. A renewal cycle is defined as the interval between two consecutive replacements or the interval between installment and the first replacement (Peng et al, 2010; Liu and Huang, 2010). In the present model, the cost items include inspection cost, downtime cost and replacement cost. Inspection cost occurs each time an inspection is carried out and downtime cost is incurred when the system is operating with the existence of failure (Berrade et al, 2012). Let K1 be the number of inspections before a failure occurs. We have P( K1 i) P Tf [iT ,(i 1)T ] R iT R (i 1)T
interval. The expected value of K1 can be obtained as
i 0
i 1
E[ K1 ] iP( K1 i ) R(iT )
E[ K 2 ] iP K 2 =i i 1
i R k1T R (k1 1)T k1 0 i 1 i 1 Pd (k1T iT ) 1 Pd (k1T jT ) j 0
(8)
As failure can only be detected at inspection and the system is replaced when a failure is discovered, the length of a renewal cycle is equal to the time interval from beginning to the last inspection. Therefore, we can obtain the expected length of a renewal as E[S ] E[ K1 ] E[ K 2 ]T
(9)
(2)
where T f is the time to failure and T is the inspection
(7)
k1 0
Denote Td as the system downtime, time interval between failure occurrence and system replacement. Expectation of Td can be expressed as E[Td ] E[S ] E[Tf ] . The expected cost within a renewal cycle is given as
E[C ] CR CI E[ K1 K2 ] Cd E[Td ]
(3)
(10)
where CR is the replacement cost, CI is the inspection cost, and Cd is the downtime cost per unit time.
1378
IFAC MIM 2016 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
Combining (9) and (10), we can have the expected long run cost rate as C (t ) E[C ] CR(T ) lim t t E[ S ] CR Cd E[T f ] C Cd I T E[ K1 K 2 ] T
CI CR Cd tdR (t ) / (11) 0 T i R k1T R (k1 1)T k1 0 T R(iT ) i 1 i 1 i 1 Pd (k1T iT ) 1 Pd (k1T jT ) j 0 Cd
4. OPTIMIZATION ALGORITHM Our goal is to determine the optimal T that minimizes the objective function CR(T), i.e.,
T arg min CR(T ) T
(12)
Analytical solution of CR(T) is hard to obtain due to the fact that E[ K1 K2 ] depends on both the degradation process and inspection interval T. Hence, we resort to Monte Carlo Simulation to calculate CR(T) (Huynh et al., 2012a). When the number of simulation histories Nl is large enough, (11) can be rewritten as Nl
Nl
n 1
n 1
CR(T ) C ( n ) / S ( n )
Combining (14) and (15), we have
E[T f ]
mT T E[ K1 K 2 ] T E[T f ] m 1 1
(16)
The proof can be concluded by substituting (16) into (11). Proposition 1 is useful in engineering, as engineers are more interested in the bounds rather than the exact value. Although the analytical solution of CR(T) is difficult to get, the upper and lower bounds of CR(T) can be obtained as an analytical solution. In situations where computational efficiency is the main concern, the bounds can be adopted instead of the exact value. Define CR Cd E[T f ] C f1 (T ) Cd I T mT E[T f ] 1 m and C CR Cd E[T f ] f 2 (T ) Cd I T T E[T f ] 1 More properties of CR(T) can be concluded as follows: Corollary. CR(T) converges to Cd for T . Proof. Based on the definition of f1 (T ) and f 2 (T ) , it is straightforward
to
see
that
lim f1 (T ) Cd
T
and
lim f 2 (T ) Cd . According to Proposition 1, we can have
T
(13)
where C(n) and S(n) are respectively the cost and length of a renewal cycle for the nth simulation history. In what follows, we design a search algorithm to determine the optimal T and CR(T).
lim CR(T ) Cd .
T
Intuitively, the corollary can be interpreted as follows: if T , then no inspection is performed for the system. As a result, the system is left failed and the long run cost rate approaches Cd .
Assume CR Cd E[Tf ] and CI Cd E[Tf ] . This assumption is reasonable in reality due to the fact that unexpected failures incur higher cost than replacement and inspection (Huynh et al, 2012a; Van Oosterom et al, 2014). Several properties of the objective function are derived for optimization algorithm. The following proposition gives the upper bound and lower bound of the objective function. Proposition 1. The long run cost rate CR(T) is limited in the interval as CR Cd E[T f ] C C CR Cd E[T f ] CR(T ) Cd I Cd I m T T E[T ] T T E[T f ] f m 1 1 Proof. K1 can be rewritten as
Tf Tf 1 K1 (14) T T T As K2 is inversely related to the number of failures at inspection and the number of failures is limited as 1 N f m , we can obtain Tf
1 1 E[ K 2 ] 1 1 m
1379
(15)
Lemma.
f1 (T ) and
f 2 (T ) are unimodel functions for
T 0, . Proof. Rewrite f 2 (T ) as a b T cT d 1 , d E[T f ] . We can where a CI , b Cd E[T f ] CR , c 1 f 2 (T ) Cd
obtain the derivative of f 2 (T ) as f 2' (T )
a bc T 2 cT d 2
Let f 2' (T ) 0 , we have
bc ac T 2
2
2acdT ad 2 0
According to the assumption that b ac , we can conclude that only one root exists over the range of 0, . Denote the root as Tu . For T 0, Tu , f 2 (T ) is monotonously
1379
IFAC MIM 2016 1380 June 28-30, 2016. Troyes, France
decreasing;
then
for
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
T Tu , ,
f 2 (T )
increases
monotonously. Combined with lim f 2 (T ) , it can be T 0
concluded that f 2 (T ) is a unimodel function for T 0, . Similarly, we can conclude f1 (T ) is a unimodel function for
T 0, . The unimodality of f1 (T ) and f 2 (T ) for T 0, is important as the property helps to determine the range of the optimum inspection interval T. Let f1 denote the minimum 2
of f1 (T ) and f be the minimum of f 2 (T ) . We can have the following proposition to determine the bounds of the optimum T. Proposition 2. The optimal inspection interval T lies in the range as T Ta , Tb , where Ta and Tb are the roots of the equation f1 (T ) f 2 ,
Ta , Tb arg f1 (T ) f 2 . T
Proof. First, we need to prove that such Ta and Tb exist. Proposition 1 shows that f1 (T ) f 2 (T ), T (0, ) . Hence, we have f1 f 2 . Also, lim f1 (T ) f 2 and T 0
lim f1 (T ) Cd f 2 . Along with the unimodality of f1 (T ) ,
T
5. AN ILLUSTRATIVE EXAMPLE A system with two fatigue crack positions is considered as an example to illustrate the proposed inspection policy. The original measurements of fatigue crack sizes are collected from Pan and Balakrishnan (2011). The data has been used for degradation modeling and reliability estimation in Pan et al (2013). In this paper, we use the same data to estimate the parameters of degradation process and further make inspection decisions. Pan et al (2013) pointed out that the aforementioned degradation process is not linearly increasing in term of time unit. A bivariate Wiener process with an exponential time
transform t is used to model the deterioration process (2013). The parameter is set to be 1.3 (Pan et al, 2013). We fit the data for the degradation model and estimate the other parameters by maximum likelihood estimation (MLE). The parameters are estimated as 1 12.2187 , 2 7.5967 , [0.0505,0.0147;0.0147,0.0247] . The parameters are used as true values in this study. The system fails if any of the two cracks exceeds 1.6 in. in size. The initial degradation levels of the two cracks are 0.9 respectively. To neutralize the effect of initial degradation, the failure threshold is set as H1 H 2 0.7 . By use of the estimates of model parameters, system reliability can be calculated based on (1). Fig. 1 shows the variation of system reliability in terms of .
we can conclude that there exist Ta and Tb that satisfy
Ta , Tb arg f1 (T ) f 2 . Let f 2 (T ) reaches the minimum at T
T Tu . The result can be proved by contradiction. Suppose T were larger than Tb , T Tb , it could be shown that CR(T ) f1 (T ) f1 (Tb ) f 2 . However, according to Proposition 1, we have CR(T ) CR(Tu ) f 2 (Tu ) f 2 , which is a contradiction. Therefore, it can be concluded that T Tb . Similarly, we can conclude that T Ta . Proposition 2 gives the lower and upper bounds of the search algorithm. Base on Proposition 2, we can design the optimization algorithm as follows. Algorithm: Optimization procedure Step 1: start from T=Ta. Step 2: Monte Carlo simulation to calculate the long-run cost rate. 2.1: initialization: set C = 0 and S = 0. 2.2: generate a history of degradation levels X(t). 2.3: compute the cost and length of a renewal cycle, C(n) a nd S(n). 2.4: compute C = C + C(n), S = S + S(n), and CR = C/S. If CR converges, go to step 3; otherwise, repeat steps 2.1-2.4. Step 3: if T
Fig. 1 Plot of system reliability when varies The cost parameters are set as CI 2 , CR 50 and Cd 5000 . The probability that a failure cannot be detected by an inspection is related to the inspection technique. In this study, is set to be 0.5. According to Proposition 2, we obtain the lower and upper bound of the optimal inspection interval as Ta 0.0021 and Tb 0.0527 . By searching within the range Ta , Tb , the minimum expected cost rate is obtained as CR 1512 , at the optimum inspection interval T 4.4 103 . Fig. 2 shows the variation of expected cost rate at different inspection intervals.
1380
IFAC MIM 2016 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
1381
appropriate inspection technique and frequency can be determined. Future research can be conducted from the following two aspects. First, in this paper, the detection probability is assumed to be constant. However, in reliability, the detection probability is not only related to inspection techniques, but also to the degradation level. It would be more reasonable to model the detection probability as a function of the degradation level. Second, other inspection policies (e.g., non-periodic inspection) are of interest to investigate. REFERENCES Fig. 2 Plot of cost rate for different inspection intervals
Sensitivity analysis is performed on in order to study the effect of different inspection techniques on the expected cost rate. Fig. 3 plots the variation of CR(T ) and T for different values of . As shown in Fig. 3, the minimum cost rate increases from 1285 to 2186 when varies from 0.1 to 0.9. The result implies that inspection accuracy has a significant impact on the cost rate. Engineers or managers are suggested to pay more efforts to improve the inspection quality so as to reduce the cost generated by hidden failures.
Fig. 3 Sensitivity analysis of CR(T ) and T on 6. CONCLUSIONS In this study, an inspection policy is proposed for systems subject to multiple degradation processes. The inspection is assumed to be imperfect, that a failure can only be discovered at inspection with certain probability. System reliability is analyzed, followed by a cost model as the objective function. Optimal inspection interval is determined by minimizing the long run cost rate. The result suggests that engineers or managers should invest on the improvement of inspection techniques in order to reduce the operation cost. This study can be applied in many practical problems, for example, to detect the crack size in railway tracks or bridges, engineers have many options such as ultrasonic testing, visual inspection and magnetic particle inspection. It is desirable to determine the most proper inspection technique and the associated inspection frequency. By balancing the cost and benefits with various testing options, the most
Barker, C. T., and Newby, M. J. (2009). Optimal nonperiodic inspection for a multivariate degradation model. Reliability Engineering and System Safety, 94(1), 33-43. Berrade, M. D. (2012). A two-phase inspection policy with imperfect testing. Applied Mathematical Modelling, 36(1), 108-114. Berrade, M. D., Cavalcante, C. A., and Scarf, P. A. (2012). Maintenance scheduling of a protection system subject to imperfect inspection and replacement. European Journal of Operational Research, 218(3), 716-725. Berrade, M. D., Scarf, P. A., Cavalcante, C. A. V., and Dwight, R. A. (2013). Imperfect inspection and replacement of a system with a defective state: A cost and reliability analysis. Reliability Engineering and System Safety, 120, 80-87. Huynh, K. T., Barros, A., and Berenguer, C. (2012a). Maintenance decision-making for systems operating under indirect condition monitoring: value of online information and impact of measurement uncertainty. IEEE Transactions on Reliability, 61(2), 410-425. Huynh, K. T., Castro, I. T., Barros, A., and Berenguer, C. (2012b). Modeling age-based maintenance strategies with minimal repairs for systems subject to competing failure modes due to degradation and shocks. European Journal of Operational Research, 218(1), 140-151. Li, W., and Pham, H. (2005). An inspection-maintenance model for systems with multiple competing processes. IEEE Transactions on Reliability, 54(2), 318-327. Liu, B., Xu, Z., Xie, M., and Kuo, W. (2014). A Value-based Preventive Maintenance Policy for Multi-component System with Continuously Degrading Components. Reliability Engineering and System Safety, 132, 83-89. Liu, X., Li, J., Al-Khalifa, K. N., Hamouda, A. S., Coit, D. W., and Elsayed, E. A. (2013). Condition-based maintenance for continuously monitored degrading systems with multiple failure modes. IIE Transactions, 45(4), 422-435. Liu, Y., and Huang, H. Z. (2010). Optimal replacement policy for multi-state system under imperfect maintenance. IEEE Transactions on Reliability, 59(3), 483-495. Pan, Z., and Balakrishnan, N. (2011). Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes. Reliability Engineering and System Safety, 96(8), 949-957. Pan, Z., Balakrishnan, N., Sun, Q., and Zhou, J. (2013). Bivariate degradation analysis of products based on
1381
IFAC MIM 2016 1382 June 28-30, 2016. Troyes, France
Bin Liu et al. / IFAC-PapersOnLine 49-12 (2016) 1377–1382
Wiener processes and copulas. Journal of Statistical Computation and Simulation, 83(7), 1316-1329. Peng, H., Feng, Q., and Coit, D. W. (2010). Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes. IIE transactions, 43(1), 12-22. Van Oosterom, C. D., Elwany, A. H., Çelebi, D., and van Houtum, G. J. (2014). Optimal policies for a delay time model with postponed replacement. European Journal of Operational Research, 232(1), 186-197. Wang, X., Balakrishnan, N., and Guo, B. (2014). Residual life estimation based on a generalized Wiener degradation process. Reliability Engineering and System Safety, 124, 13-23. Wang, Y., and Pham, H. (2012). Modeling the dependent competing risks with multiple degradation processes and random shock using time-varying copulas. IEEE Transactions on Reliability, 61(1), 13-22. Ye, Z. S., and Xie, M. (2015). Stochastic modelling and analysis of degradation for highly reliable products. Applied Stochastic Models in Business and Industry, 31(1), 16-32. Ye, Z. S., Xie, M., Tang, L. C., and Shen, Y. (2012). Degradation-based burn-in planning under competing risks. Technometrics, 54(2), 159-168.
1382