Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 947104 9 pageshttpdxdoiorg1011552013947104
Research ArticleRisk-Based Predictive Maintenance for Safety-CriticalSystems by Using Probabilistic Inference
Tianhua Xu1 Tao Tang1 Haifeng Wang2 and Tangming Yuan3
1 State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China2National Engineering Research Centre of Rail Transportation Operation and Control Systems Beijing Jiaotong UniversityBeijing 100044 China
3 Computer Science Department University of York York YO10 5GH UK
Correspondence should be addressed to Tianhua Xu thxubjtugmailcom
Received 27 January 2013 Revised 13 June 2013 Accepted 27 June 2013
Academic Editor Suiyang Khoo
Copyright copy 2013 Tianhua Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Risk-based maintenance (RBM) aims to improve maintenance planning and decision making by reducing the probability andconsequences of failure of equipment A new predictive maintenance strategy that integrates dynamic evolution model and riskassessment is proposed which can be used to calculate the optimal maintenance time with minimal cost and safety constraintsThedynamic evolutionmodel provides qualified risks by using probabilistic inference with bucket elimination and gives the prospectivedegradation trend of a complex system Based on the degradation trend an optimal maintenance time can be determined byminimizing the expected maintenance cost per time unit The effectiveness of the proposed method is validated and demonstratedby a collision accident of high-speed trains with obstacles in the presence of safety and cost constrains
1 Introduction
Safety-critical systems such as chemical factory nuclearplant and train control systems are those where failurescould result in loss of life significant property damage ordamage to the environmentThe loss caused by safety criticalsystem failures is now becoming difficult to estimate Theefficient maintenance strategies are playing more importantroles in preventing such system failures
Over the last decade reactive (fixing or replacing equip-ment after it fails) or blindly proactive strategies (also knownas preventive strategies) have been used for system main-tenance The main disadvantage of both approaches is thatthey are extremely wasteful As condition-basedmaintenance(CBM) systems have been implemented in a way to contin-uously output data that is calculated against the status andperformance of the equipment the decision making in CBMfocuses on predictive maintenance (PdM) which promisesto reduce downtime spare inventory maintenance cost andsafety hazards Much work has been carried out in the area ofpredictive maintenance in order to improve safety Generally
speaking current prognostic approaches can be classifiedinto three categoriesnamely model-based data-driven andhybrid prognostics [1] For example in [2] a DBN-HAZOPmodel was proposed to deduce the opportunistic predictivemaintenance for complex multicomponent systems The keyidea behind the model is reliability-based maintenanceKrishnasamy et al [3] proposed the risk-based mainte-nance (RBM) methodology and a case study of a power-generating unit was used to illustrate the methodologyArunraj and Maiti [4] identified the risk analysis and risk-based maintenance methodologies and classified them intosuitable classes
With the aforementioned research contributed to theefficient maintenance strategies of systems to the best ofour knowledge however the integration of dynamic systemfailure scenario into a risk-based maintenance model andthe adoption of efficient inference methods for optimalmaintenance strategy have received little attention For theintegration of dynamic system failure scenario with risk-based maintenance model reference [2] proposed a com-ponent dynamic failure for reliability-based maintenance
2 Mathematical Problems in Engineering
model In themodel the key focus is on reliability rather thanrisk Regarding the inference approaches to manipulating themaintenance model junction tree algorithms [5 6] are quitecommon and widely used However junction algorithms arecomparatively complex which demands long digressions ongraph theoretic concepts Although there has been effortto explain junction tree algorithms without resorting tographical concepts [7] the effort has not produced a variableelimination-like scheme for inference
In order to tackle these problems we propose a 2-TBN(two-slice temporal Bayes net) and risk-based maintenancemodel By encoding the failure scenario into the condi-tional probability table (CPT) of risk based maintenancemodel the risk of failure scenario is embedded In order tofacilitate efficient inference an ad hoc bucket-elimination-based probabilistic inference is presented Comparing withthe complex junction-tree based inference an attractiveproperty of bucket elimination approaches is that it isrelatively easy to understand and implement Finally byutilizing the optimal theory the optimal maintenance timeinterval with minimal cost and risk constraints can beobtained
The rest of the paper is organized as follows In Section 2the principle of RBM and the proposed RBM methodol-ogy are introduced In Section 3 a maintenance model fordegradation and risk prediction is presented Section 4 givesoptimal predictive maintenance strategies A case study ofa collision between a high speed train and an obstacle isdiscussed in Section 5 Section 6 draws the conclusion of thepaper
2 Risk-Based Maintenance(RBM) Methodology
Risk-based maintenance methodology provides a tool formaintenance planning and decision making to reduce theprobability and consequences of failure of equipment Theresulting maintenance program minimizes the risk of thesystem and the maintenance cost Figure 1 shows a generalfollow diagram of RBM It consists of the following steps(1) identification of components subsystems system and theirrelationships the system is divided into subsystems and thecomponents of each subsystem and their relationships areidentified in the following sections we model the systemstructure by using a special case of dynamic Bayesian net-work the 2-TBN (2)Collecting failure data failure modeland failure rate the information is encoded in the CPT in2-TBNg based maintenance model(3) Risk assessment andevaluation by using probabilistic inference with bucket elim-ination a consequence analysis is implemented to quantifythe effect of the occurrence of each failure scenario andobtain quantitativemeasure for its associated risksThe risk isused to study maintenance costs including the costs incurredas a result of failure (4) Optimal maintenance strategy bydefining different maintenance costs the optimal mainte-nance scheme can be derived by applying the optimizationtheory to the risk quantitative measure computed in theaforementioned step
3 Maintenance Model forDegradation and Risk Prediction
This section illustrates the first two steps of the RBMarchitecture discussed in Section 2 above The main purposeis to encode the states the dependency relations amongcomponents in each subsystem subsystems and the systemIn order to facilitate the understanding of the optimizationof predictive maintenance we first introduce some basicnotions including dynamic Bayesian network (DBN) and2-TBN Model We then prescribe the maintenance modelbased on 2-TBN and discuss the opportunistic predictivemaintenance strategies
31 Dynamic Bayesian Network and 2-TBN Model A Baye-sian network (BN) is a directed acyclic graph (DAG) whichis a probability-based knowledge representation methodand appropriate for the modeling of causal processes withuncertainty The formal notion is defined as follows
Definition 1 (see [8]) A Bayesian network (BN) is a triple(119881 119866 119875) where 119881 is a set of variables 119866 is a connecteddirected acyclic graph (DAG) and there is a one-to-onecorrespondence between nodes in 119866 and variables in 119881 119875 isa set of probability distribution 119875 = 119875(V | 120587(V)) | V isin 119881where 120587(V) denotes the set of parents of V in 119866
The statistic Bayesian network can be extended to adynamic Bayesian network (DBN) by introducing relevanttemporal dependences that capture the dynamic behaviors ofthe domain variables at different times of a static networkDefinition 2 gives the formal definition of DBN
Definition 2 A dynamic Bayesian network (DBN) is aquadruplet 119866 = (⋃
119905=0119881119905 ⋃119905=0119864119905 ⋃119905=0119864rarr119905 ⋃119905=0119875119905) and
each119881119905is a set of nodes labeled by variables which represents
the dynamic domain at time instant 119905 (0 le 119905 lt 119896) Collec-tively⋃119896
119905=0119881119905represents the dynamic domain over 119896 instants
Each 119864119905is a set of arcs among nodes in 119881
119905 which represents
dependencies among domain variables at time 119905 Each 119864rarr119905
isa set of temporal arcs each of which is directed from a nodein 119881119905minus1
to a node in 119881119905(0 lt 119905 lt 119896) 119875
119905is set of probability
distributions which can be referred to [8]
In this paper we only consider a special class of DBNswhich is called 2-slice temporal Bayesian network (2-TBN)[9] A 2-TBN is a DBN which satisfies the Markov propertyof order 1 that is the future is independent of its past givenits present
32 2-TBN Based Maintenance Model 2-TBNs are generaltools allowing the modeling of dynamic complex systemsBesides it is important to note that using 2-TBNs to representa variable depending on its own past is equivalent to theuse of Markov chain to describe its local transition modelConsequently we propose a 2-TBN based maintenancemodel capable of representing dynamic degradation and risklevel of subsystems We treat system state system failureand accidents as random variables and model dependencies
maintenance policy toreduce the risk level toan acceptable level and
minimize the repair cost
Figure 1 Architecture of RBMmethodology (revised after [3])
middot middot middot
middot middot middot
RC1
X1
1Y1
1
F1
1
X1
2Y1
2
F1
2
X1
NY1
N
F1
N
(a) Initial model
middot middot middot
middot middot middotmiddot middot middot
middot middot middot
RCtminus1 RCt
Xtminus1
1Ytminus1
1
Ftminus1
1
Xtminus1
2 Ytminus1
2
Ftminus1
2
Xtminus1
NYtminus1
N
Ftminus1
N
Xt
1Yt
1
Ft
1
Xt
2Yt
2
Ft
2
Xt
NYt
N
Ft
N
(b) Transition model
Figure 2 2-TBN basedmaintenancemodel (a) gives the initial state of themaintenancemodel while (b) depicts its transitionmodel betweentime slice 119905 minus 1 and 119905
among them by exploiting the use of conditional probabilitytables (CPT) In order to simplify calculation all of variablesin our model are assumed to be discrete
The 2-TBN based maintenance model is depicted inFigure 2 The model consists of the following variables119883119905
119894(1 le 119894 le 119873 1 le 119905) the state of component 119883 (eg failure
or ok) in the 119894th subsystem at time instant 119905 (for the sake ofsimplicity only two components 119883 and 119884 are shown in thefigure) 119865119905
119894(1 le 119894 le 119873 1 le 119905) denotes the states of 119894th
subsystem (eg failure or ok) at time instant 119905 119860119905 (1 le 119905)denotes the accident probability of the system due to thesubsystem failure RC119905 (1 le 119905) represents the corresponding
4 Mathematical Problems in Engineering
Table 1 Conditional probabilities for component state 1198831 1198841 and
subsystem 1198651
119883119905
1119884119905
1Pr(1198651199051| 119883119905
1 119884119905
1)
Ok Ok OkOk Fail FailFail Ok FailFail Fail Fail
Table 2 Temporal CPT for component1198831
119883119905minus1119894
119883119905119894
Pr(119883119905119894| 119883119905minus1119894)
Ok Ok 1 minus 120582119901Δ119879
Ok Fail 120582119901Δ119879
Fail Ok 0Fail Fail 1
maintenance cost till time instant 119905 From Figure 2(b) itcan be seen that the current component state for example119883119905
119894depends on the previous component states for example
119883119905minus1119894
To complete the 2-TBN based maintenance model the
conditional probabilities must be specified for (1) the statetransition of components between different time slice and(2) the dependency of components output on the subsystemsystem and accident state For example assume the state ofcomponent 119883119905
119894(1 le 119894 le 119873 1 le 119905) has only two values ok
or fail then its dependencies among 119865119905119894(1 le 119894 le 119873 1 le 119905)
and 119883119905119894 119884119905119894(1 le 119894 le 119873 1 le 119905) can be illustrated by the CPT
as shown in Table 1 In other words any failure in component1198831andor 119884
1will lead to the failure of subsystem 119865
1
Similarly under the assumption that the failure rate ofa component follows an exponential distribution where allthese transition rates are constant the transition relationsbetween consecutive nodes for the different componentsmaintenance model are obtained as follows (the failure rate isdenoted by 120582
119894 the time interval between two successive trials
is denoted byΔ119879 and the components are assumed to be newon the initial trial 119905 = 0)
(Pr (119883119905119894= 0)
Pr (119883119905119894= 1)
) = (119890minus120582119896Δ119879 0
1 minus 119890minus120582119896Δ119879 1)(
Pr (119883119905minus1119894= 0)
Pr (119883119905minus1119894= 1)
) (1)
So the conditional probabilities for state transitions can beobtained directly from the above equation For examplePr(119883119905+1119894
= 119886 | 119883119905119894= 0) can be obtained where 119886 denotes
normal or failure state
(Pr (119883119905+1
119894= 0 | 119883119905
119894= 0)
Pr (119883119905+1119894= 1 | 119883119905
119894= 0)
) = (119890minus120582119896Δ119879
1 minus 119890minus120582119896Δ119879) (2)
The corresponding temporal CPT for component119883119905119894(1 le 119894 le
119873 1 le 119905) is obtained as shown in Table 2Finally the consequence resulting from different sub-
system failures (ie 119865119905119894(1 le 119894 le 119873 1 le 119905)) can be
classified as shown in Table 3 The specific consequence canbe determined by different failure remain so manually
Table 3 Consequence resulted from different subsystem failurescenario
119865119905
1119865119905
2sdot sdot sdot 119865
119905
119873Consequence
Ok Ok sdot sdot sdot Ok NoFail Ok sdot sdot sdot Ok InsignificantFail Fail sdot sdot sdot Ok Marginal (minor injury)Fail Fail sdot sdot sdot Ok Critical (single severe injury)Fail Fail sdot sdot sdot Fail Catastrophic (fatalities)
The risk can be computed by integration of consequenceand probability resulting from different failure scenariosPlease note that the probability of subsystem failure scenariocan be calculated by using probability inference from the 2-TBN based maintenance model The detailed procedures arediscussed in the following sections
4 Optimal Predictive Maintenance Strategies
This section discusses the calculation of optimal predictivemaintenance strategies which consists of the calculation ofthe failure and accident probability of themaintenancemodeland optimal maintenance time under the repairing costconstraints
41 Calculation of the Failure Probability of a Componentin Maintenance Model The purpose of this subsection is toevaluate the probability of any failure scenario for a timelength of 119879 In other words the underlying problem boilsdown to the calculation of the following probability
Ψ119905= Pr (1198651 119865119879) (3)
The following theorem gives a recursive characterization ofΨ119905 based on the derivation of the bucket elimination methodpresented in [10]
Theorem 3 (recursive characterization of Ψ119905) Let (1198831199051
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
model In themodel the key focus is on reliability rather thanrisk Regarding the inference approaches to manipulating themaintenance model junction tree algorithms [5 6] are quitecommon and widely used However junction algorithms arecomparatively complex which demands long digressions ongraph theoretic concepts Although there has been effortto explain junction tree algorithms without resorting tographical concepts [7] the effort has not produced a variableelimination-like scheme for inference
In order to tackle these problems we propose a 2-TBN(two-slice temporal Bayes net) and risk-based maintenancemodel By encoding the failure scenario into the condi-tional probability table (CPT) of risk based maintenancemodel the risk of failure scenario is embedded In order tofacilitate efficient inference an ad hoc bucket-elimination-based probabilistic inference is presented Comparing withthe complex junction-tree based inference an attractiveproperty of bucket elimination approaches is that it isrelatively easy to understand and implement Finally byutilizing the optimal theory the optimal maintenance timeinterval with minimal cost and risk constraints can beobtained
The rest of the paper is organized as follows In Section 2the principle of RBM and the proposed RBM methodol-ogy are introduced In Section 3 a maintenance model fordegradation and risk prediction is presented Section 4 givesoptimal predictive maintenance strategies A case study ofa collision between a high speed train and an obstacle isdiscussed in Section 5 Section 6 draws the conclusion of thepaper
2 Risk-Based Maintenance(RBM) Methodology
Risk-based maintenance methodology provides a tool formaintenance planning and decision making to reduce theprobability and consequences of failure of equipment Theresulting maintenance program minimizes the risk of thesystem and the maintenance cost Figure 1 shows a generalfollow diagram of RBM It consists of the following steps(1) identification of components subsystems system and theirrelationships the system is divided into subsystems and thecomponents of each subsystem and their relationships areidentified in the following sections we model the systemstructure by using a special case of dynamic Bayesian net-work the 2-TBN (2)Collecting failure data failure modeland failure rate the information is encoded in the CPT in2-TBNg based maintenance model(3) Risk assessment andevaluation by using probabilistic inference with bucket elim-ination a consequence analysis is implemented to quantifythe effect of the occurrence of each failure scenario andobtain quantitativemeasure for its associated risksThe risk isused to study maintenance costs including the costs incurredas a result of failure (4) Optimal maintenance strategy bydefining different maintenance costs the optimal mainte-nance scheme can be derived by applying the optimizationtheory to the risk quantitative measure computed in theaforementioned step
3 Maintenance Model forDegradation and Risk Prediction
This section illustrates the first two steps of the RBMarchitecture discussed in Section 2 above The main purposeis to encode the states the dependency relations amongcomponents in each subsystem subsystems and the systemIn order to facilitate the understanding of the optimizationof predictive maintenance we first introduce some basicnotions including dynamic Bayesian network (DBN) and2-TBN Model We then prescribe the maintenance modelbased on 2-TBN and discuss the opportunistic predictivemaintenance strategies
31 Dynamic Bayesian Network and 2-TBN Model A Baye-sian network (BN) is a directed acyclic graph (DAG) whichis a probability-based knowledge representation methodand appropriate for the modeling of causal processes withuncertainty The formal notion is defined as follows
Definition 1 (see [8]) A Bayesian network (BN) is a triple(119881 119866 119875) where 119881 is a set of variables 119866 is a connecteddirected acyclic graph (DAG) and there is a one-to-onecorrespondence between nodes in 119866 and variables in 119881 119875 isa set of probability distribution 119875 = 119875(V | 120587(V)) | V isin 119881where 120587(V) denotes the set of parents of V in 119866
The statistic Bayesian network can be extended to adynamic Bayesian network (DBN) by introducing relevanttemporal dependences that capture the dynamic behaviors ofthe domain variables at different times of a static networkDefinition 2 gives the formal definition of DBN
Definition 2 A dynamic Bayesian network (DBN) is aquadruplet 119866 = (⋃
119905=0119881119905 ⋃119905=0119864119905 ⋃119905=0119864rarr119905 ⋃119905=0119875119905) and
each119881119905is a set of nodes labeled by variables which represents
the dynamic domain at time instant 119905 (0 le 119905 lt 119896) Collec-tively⋃119896
119905=0119881119905represents the dynamic domain over 119896 instants
Each 119864119905is a set of arcs among nodes in 119881
119905 which represents
dependencies among domain variables at time 119905 Each 119864rarr119905
isa set of temporal arcs each of which is directed from a nodein 119881119905minus1
to a node in 119881119905(0 lt 119905 lt 119896) 119875
119905is set of probability
distributions which can be referred to [8]
In this paper we only consider a special class of DBNswhich is called 2-slice temporal Bayesian network (2-TBN)[9] A 2-TBN is a DBN which satisfies the Markov propertyof order 1 that is the future is independent of its past givenits present
32 2-TBN Based Maintenance Model 2-TBNs are generaltools allowing the modeling of dynamic complex systemsBesides it is important to note that using 2-TBNs to representa variable depending on its own past is equivalent to theuse of Markov chain to describe its local transition modelConsequently we propose a 2-TBN based maintenancemodel capable of representing dynamic degradation and risklevel of subsystems We treat system state system failureand accidents as random variables and model dependencies
maintenance policy toreduce the risk level toan acceptable level and
minimize the repair cost
Figure 1 Architecture of RBMmethodology (revised after [3])
middot middot middot
middot middot middot
RC1
X1
1Y1
1
F1
1
X1
2Y1
2
F1
2
X1
NY1
N
F1
N
(a) Initial model
middot middot middot
middot middot middotmiddot middot middot
middot middot middot
RCtminus1 RCt
Xtminus1
1Ytminus1
1
Ftminus1
1
Xtminus1
2 Ytminus1
2
Ftminus1
2
Xtminus1
NYtminus1
N
Ftminus1
N
Xt
1Yt
1
Ft
1
Xt
2Yt
2
Ft
2
Xt
NYt
N
Ft
N
(b) Transition model
Figure 2 2-TBN basedmaintenancemodel (a) gives the initial state of themaintenancemodel while (b) depicts its transitionmodel betweentime slice 119905 minus 1 and 119905
among them by exploiting the use of conditional probabilitytables (CPT) In order to simplify calculation all of variablesin our model are assumed to be discrete
The 2-TBN based maintenance model is depicted inFigure 2 The model consists of the following variables119883119905
119894(1 le 119894 le 119873 1 le 119905) the state of component 119883 (eg failure
or ok) in the 119894th subsystem at time instant 119905 (for the sake ofsimplicity only two components 119883 and 119884 are shown in thefigure) 119865119905
119894(1 le 119894 le 119873 1 le 119905) denotes the states of 119894th
subsystem (eg failure or ok) at time instant 119905 119860119905 (1 le 119905)denotes the accident probability of the system due to thesubsystem failure RC119905 (1 le 119905) represents the corresponding
4 Mathematical Problems in Engineering
Table 1 Conditional probabilities for component state 1198831 1198841 and
subsystem 1198651
119883119905
1119884119905
1Pr(1198651199051| 119883119905
1 119884119905
1)
Ok Ok OkOk Fail FailFail Ok FailFail Fail Fail
Table 2 Temporal CPT for component1198831
119883119905minus1119894
119883119905119894
Pr(119883119905119894| 119883119905minus1119894)
Ok Ok 1 minus 120582119901Δ119879
Ok Fail 120582119901Δ119879
Fail Ok 0Fail Fail 1
maintenance cost till time instant 119905 From Figure 2(b) itcan be seen that the current component state for example119883119905
119894depends on the previous component states for example
119883119905minus1119894
To complete the 2-TBN based maintenance model the
conditional probabilities must be specified for (1) the statetransition of components between different time slice and(2) the dependency of components output on the subsystemsystem and accident state For example assume the state ofcomponent 119883119905
119894(1 le 119894 le 119873 1 le 119905) has only two values ok
or fail then its dependencies among 119865119905119894(1 le 119894 le 119873 1 le 119905)
and 119883119905119894 119884119905119894(1 le 119894 le 119873 1 le 119905) can be illustrated by the CPT
as shown in Table 1 In other words any failure in component1198831andor 119884
1will lead to the failure of subsystem 119865
1
Similarly under the assumption that the failure rate ofa component follows an exponential distribution where allthese transition rates are constant the transition relationsbetween consecutive nodes for the different componentsmaintenance model are obtained as follows (the failure rate isdenoted by 120582
119894 the time interval between two successive trials
is denoted byΔ119879 and the components are assumed to be newon the initial trial 119905 = 0)
(Pr (119883119905119894= 0)
Pr (119883119905119894= 1)
) = (119890minus120582119896Δ119879 0
1 minus 119890minus120582119896Δ119879 1)(
Pr (119883119905minus1119894= 0)
Pr (119883119905minus1119894= 1)
) (1)
So the conditional probabilities for state transitions can beobtained directly from the above equation For examplePr(119883119905+1119894
= 119886 | 119883119905119894= 0) can be obtained where 119886 denotes
normal or failure state
(Pr (119883119905+1
119894= 0 | 119883119905
119894= 0)
Pr (119883119905+1119894= 1 | 119883119905
119894= 0)
) = (119890minus120582119896Δ119879
1 minus 119890minus120582119896Δ119879) (2)
The corresponding temporal CPT for component119883119905119894(1 le 119894 le
119873 1 le 119905) is obtained as shown in Table 2Finally the consequence resulting from different sub-
system failures (ie 119865119905119894(1 le 119894 le 119873 1 le 119905)) can be
classified as shown in Table 3 The specific consequence canbe determined by different failure remain so manually
Table 3 Consequence resulted from different subsystem failurescenario
119865119905
1119865119905
2sdot sdot sdot 119865
119905
119873Consequence
Ok Ok sdot sdot sdot Ok NoFail Ok sdot sdot sdot Ok InsignificantFail Fail sdot sdot sdot Ok Marginal (minor injury)Fail Fail sdot sdot sdot Ok Critical (single severe injury)Fail Fail sdot sdot sdot Fail Catastrophic (fatalities)
The risk can be computed by integration of consequenceand probability resulting from different failure scenariosPlease note that the probability of subsystem failure scenariocan be calculated by using probability inference from the 2-TBN based maintenance model The detailed procedures arediscussed in the following sections
4 Optimal Predictive Maintenance Strategies
This section discusses the calculation of optimal predictivemaintenance strategies which consists of the calculation ofthe failure and accident probability of themaintenancemodeland optimal maintenance time under the repairing costconstraints
41 Calculation of the Failure Probability of a Componentin Maintenance Model The purpose of this subsection is toevaluate the probability of any failure scenario for a timelength of 119879 In other words the underlying problem boilsdown to the calculation of the following probability
Ψ119905= Pr (1198651 119865119879) (3)
The following theorem gives a recursive characterization ofΨ119905 based on the derivation of the bucket elimination methodpresented in [10]
Theorem 3 (recursive characterization of Ψ119905) Let (1198831199051
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
maintenance policy toreduce the risk level toan acceptable level and
minimize the repair cost
Figure 1 Architecture of RBMmethodology (revised after [3])
middot middot middot
middot middot middot
RC1
X1
1Y1
1
F1
1
X1
2Y1
2
F1
2
X1
NY1
N
F1
N
(a) Initial model
middot middot middot
middot middot middotmiddot middot middot
middot middot middot
RCtminus1 RCt
Xtminus1
1Ytminus1
1
Ftminus1
1
Xtminus1
2 Ytminus1
2
Ftminus1
2
Xtminus1
NYtminus1
N
Ftminus1
N
Xt
1Yt
1
Ft
1
Xt
2Yt
2
Ft
2
Xt
NYt
N
Ft
N
(b) Transition model
Figure 2 2-TBN basedmaintenancemodel (a) gives the initial state of themaintenancemodel while (b) depicts its transitionmodel betweentime slice 119905 minus 1 and 119905
among them by exploiting the use of conditional probabilitytables (CPT) In order to simplify calculation all of variablesin our model are assumed to be discrete
The 2-TBN based maintenance model is depicted inFigure 2 The model consists of the following variables119883119905
119894(1 le 119894 le 119873 1 le 119905) the state of component 119883 (eg failure
or ok) in the 119894th subsystem at time instant 119905 (for the sake ofsimplicity only two components 119883 and 119884 are shown in thefigure) 119865119905
119894(1 le 119894 le 119873 1 le 119905) denotes the states of 119894th
subsystem (eg failure or ok) at time instant 119905 119860119905 (1 le 119905)denotes the accident probability of the system due to thesubsystem failure RC119905 (1 le 119905) represents the corresponding
4 Mathematical Problems in Engineering
Table 1 Conditional probabilities for component state 1198831 1198841 and
subsystem 1198651
119883119905
1119884119905
1Pr(1198651199051| 119883119905
1 119884119905
1)
Ok Ok OkOk Fail FailFail Ok FailFail Fail Fail
Table 2 Temporal CPT for component1198831
119883119905minus1119894
119883119905119894
Pr(119883119905119894| 119883119905minus1119894)
Ok Ok 1 minus 120582119901Δ119879
Ok Fail 120582119901Δ119879
Fail Ok 0Fail Fail 1
maintenance cost till time instant 119905 From Figure 2(b) itcan be seen that the current component state for example119883119905
119894depends on the previous component states for example
119883119905minus1119894
To complete the 2-TBN based maintenance model the
conditional probabilities must be specified for (1) the statetransition of components between different time slice and(2) the dependency of components output on the subsystemsystem and accident state For example assume the state ofcomponent 119883119905
119894(1 le 119894 le 119873 1 le 119905) has only two values ok
or fail then its dependencies among 119865119905119894(1 le 119894 le 119873 1 le 119905)
and 119883119905119894 119884119905119894(1 le 119894 le 119873 1 le 119905) can be illustrated by the CPT
as shown in Table 1 In other words any failure in component1198831andor 119884
1will lead to the failure of subsystem 119865
1
Similarly under the assumption that the failure rate ofa component follows an exponential distribution where allthese transition rates are constant the transition relationsbetween consecutive nodes for the different componentsmaintenance model are obtained as follows (the failure rate isdenoted by 120582
119894 the time interval between two successive trials
is denoted byΔ119879 and the components are assumed to be newon the initial trial 119905 = 0)
(Pr (119883119905119894= 0)
Pr (119883119905119894= 1)
) = (119890minus120582119896Δ119879 0
1 minus 119890minus120582119896Δ119879 1)(
Pr (119883119905minus1119894= 0)
Pr (119883119905minus1119894= 1)
) (1)
So the conditional probabilities for state transitions can beobtained directly from the above equation For examplePr(119883119905+1119894
= 119886 | 119883119905119894= 0) can be obtained where 119886 denotes
normal or failure state
(Pr (119883119905+1
119894= 0 | 119883119905
119894= 0)
Pr (119883119905+1119894= 1 | 119883119905
119894= 0)
) = (119890minus120582119896Δ119879
1 minus 119890minus120582119896Δ119879) (2)
The corresponding temporal CPT for component119883119905119894(1 le 119894 le
119873 1 le 119905) is obtained as shown in Table 2Finally the consequence resulting from different sub-
system failures (ie 119865119905119894(1 le 119894 le 119873 1 le 119905)) can be
classified as shown in Table 3 The specific consequence canbe determined by different failure remain so manually
Table 3 Consequence resulted from different subsystem failurescenario
119865119905
1119865119905
2sdot sdot sdot 119865
119905
119873Consequence
Ok Ok sdot sdot sdot Ok NoFail Ok sdot sdot sdot Ok InsignificantFail Fail sdot sdot sdot Ok Marginal (minor injury)Fail Fail sdot sdot sdot Ok Critical (single severe injury)Fail Fail sdot sdot sdot Fail Catastrophic (fatalities)
The risk can be computed by integration of consequenceand probability resulting from different failure scenariosPlease note that the probability of subsystem failure scenariocan be calculated by using probability inference from the 2-TBN based maintenance model The detailed procedures arediscussed in the following sections
4 Optimal Predictive Maintenance Strategies
This section discusses the calculation of optimal predictivemaintenance strategies which consists of the calculation ofthe failure and accident probability of themaintenancemodeland optimal maintenance time under the repairing costconstraints
41 Calculation of the Failure Probability of a Componentin Maintenance Model The purpose of this subsection is toevaluate the probability of any failure scenario for a timelength of 119879 In other words the underlying problem boilsdown to the calculation of the following probability
Ψ119905= Pr (1198651 119865119879) (3)
The following theorem gives a recursive characterization ofΨ119905 based on the derivation of the bucket elimination methodpresented in [10]
Theorem 3 (recursive characterization of Ψ119905) Let (1198831199051
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
Table 1 Conditional probabilities for component state 1198831 1198841 and
subsystem 1198651
119883119905
1119884119905
1Pr(1198651199051| 119883119905
1 119884119905
1)
Ok Ok OkOk Fail FailFail Ok FailFail Fail Fail
Table 2 Temporal CPT for component1198831
119883119905minus1119894
119883119905119894
Pr(119883119905119894| 119883119905minus1119894)
Ok Ok 1 minus 120582119901Δ119879
Ok Fail 120582119901Δ119879
Fail Ok 0Fail Fail 1
maintenance cost till time instant 119905 From Figure 2(b) itcan be seen that the current component state for example119883119905
119894depends on the previous component states for example
119883119905minus1119894
To complete the 2-TBN based maintenance model the
conditional probabilities must be specified for (1) the statetransition of components between different time slice and(2) the dependency of components output on the subsystemsystem and accident state For example assume the state ofcomponent 119883119905
119894(1 le 119894 le 119873 1 le 119905) has only two values ok
or fail then its dependencies among 119865119905119894(1 le 119894 le 119873 1 le 119905)
and 119883119905119894 119884119905119894(1 le 119894 le 119873 1 le 119905) can be illustrated by the CPT
as shown in Table 1 In other words any failure in component1198831andor 119884
1will lead to the failure of subsystem 119865
1
Similarly under the assumption that the failure rate ofa component follows an exponential distribution where allthese transition rates are constant the transition relationsbetween consecutive nodes for the different componentsmaintenance model are obtained as follows (the failure rate isdenoted by 120582
119894 the time interval between two successive trials
is denoted byΔ119879 and the components are assumed to be newon the initial trial 119905 = 0)
(Pr (119883119905119894= 0)
Pr (119883119905119894= 1)
) = (119890minus120582119896Δ119879 0
1 minus 119890minus120582119896Δ119879 1)(
Pr (119883119905minus1119894= 0)
Pr (119883119905minus1119894= 1)
) (1)
So the conditional probabilities for state transitions can beobtained directly from the above equation For examplePr(119883119905+1119894
= 119886 | 119883119905119894= 0) can be obtained where 119886 denotes
normal or failure state
(Pr (119883119905+1
119894= 0 | 119883119905
119894= 0)
Pr (119883119905+1119894= 1 | 119883119905
119894= 0)
) = (119890minus120582119896Δ119879
1 minus 119890minus120582119896Δ119879) (2)
The corresponding temporal CPT for component119883119905119894(1 le 119894 le
119873 1 le 119905) is obtained as shown in Table 2Finally the consequence resulting from different sub-
system failures (ie 119865119905119894(1 le 119894 le 119873 1 le 119905)) can be
classified as shown in Table 3 The specific consequence canbe determined by different failure remain so manually
Table 3 Consequence resulted from different subsystem failurescenario
119865119905
1119865119905
2sdot sdot sdot 119865
119905
119873Consequence
Ok Ok sdot sdot sdot Ok NoFail Ok sdot sdot sdot Ok InsignificantFail Fail sdot sdot sdot Ok Marginal (minor injury)Fail Fail sdot sdot sdot Ok Critical (single severe injury)Fail Fail sdot sdot sdot Fail Catastrophic (fatalities)
The risk can be computed by integration of consequenceand probability resulting from different failure scenariosPlease note that the probability of subsystem failure scenariocan be calculated by using probability inference from the 2-TBN based maintenance model The detailed procedures arediscussed in the following sections
4 Optimal Predictive Maintenance Strategies
This section discusses the calculation of optimal predictivemaintenance strategies which consists of the calculation ofthe failure and accident probability of themaintenancemodeland optimal maintenance time under the repairing costconstraints
41 Calculation of the Failure Probability of a Componentin Maintenance Model The purpose of this subsection is toevaluate the probability of any failure scenario for a timelength of 119879 In other words the underlying problem boilsdown to the calculation of the following probability
Ψ119905= Pr (1198651 119865119879) (3)
The following theorem gives a recursive characterization ofΨ119905 based on the derivation of the bucket elimination methodpresented in [10]
Theorem 3 (recursive characterization of Ψ119905) Let (1198831199051
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
Then the computation of Ψ119905can be simplified as follows
Ψ119905= sum
(119883119905119894 119884119905
119894119860119905)1le119894le119873
119873
prod119894=1
[Pr (119883119905119894| 119883119905minus1
119894)Pr (119884119905
119894| 119884119905minus1
119894)]
sdot
119873
prod119895=1
Pr (119865119905119895| 119883119905
119895 119884119905
119895)
sdot Pr (119860119905 | 1198651199051 119865
119905
119873)
sdot Ψ119905minus1
(10)
So the theorem can be proved now
Remark 4 The bucket-elimination based inference approachpresented in the risk-based maintenance model aims atefficiently computing the failure probability densities ofcomponents in a maintenance model which is representedin a dynamic Bayesian network The construction of buckettree simplifies the presentation and produces an algorithmthat is easy to grasp and implement The algorithm reliesonly on independency relations and probability manipula-tion and does not use graphical concepts such as trian-gulations and cliques and it focuses solely on the prob-ability densities and avoids complex digressions on graphtheoretic concepts
42 Calculation of the Optimization Maintenance Time Thissubsection concerns the optimization of predictive main-tenance under the criterion of minimizing its life timeoperation and repair costs Similar to [2] two types of costsneed to be considered (1) the cost of repairing componentdegradation of failure which is termed as ldquorepairing costrdquoand (2) production losses caused by the shutdown of thesystem to undertake repairs which is related to the time lost inthese tasks There are two kinds of repairing costs correctiverepairing cost needs to be charged when component failureoccurs before proactive schedule time and proactive repairingcost is charged when component is under repair or replace-ment at certain proactive scheduled time without failureFor the 119894th component the specific corrective and proactiverepair costs are denoted as RC119888
119894and RC119901
119894 respectively We
consider the latter less than the former because the formercontains production loss personal injures and environment
contamination For the 119894th component the expected total costper unit time of predictive maintenance is given by
RC119903 119894(119905) =
RC119888119894119865119894(119905) + RC119901
119894(1 minus 119865
119894(119905))
119905 (11)
where 119905 is the time for a proactive repair of component 119894and 119865
119894(119905) is its failure probability distribution It represents
the cumulative distribution function of random variable 119865119894
ldquotime to failurerdquo which is the output of the 2-TBN basedmaintenance model If the system contains 119873 componentsthat is119866 = 1 2 119873 the expected group repair cost ratesare given as follows
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
Figure 4 Event tree analysis of collision accident of high-speedtrains
where 2TBNMM denotes the 2-TBN based maintenancemodel
5 Case Study
In this section an accident for a high speed train withan obstacle located on the rail segment is considered todemonstrate the feasibility and effectiveness of the proposedapproach Figure 3 shows the configuration of the accidentwhich consists of signal track circuit computer interlockingsystem and train control system Signals are placed betweentrack segments and show different aspects These aspectsinform the train driver to go or stop safely track circuitis monitored by electrical equipment to detect the presenceof a train It can also be used to send allowable trainvelocity code to assure the train moving safely Computerinterlocking system (CI) is used to give the right routefor a train to enter the station If a route is successfullyestablished CI will inform the signal to display green aspectOtherwise the red aspect will be displayed Train controlsystem receives the allowable train velocity code from thetrack circuit and the signal aspect and then determineswhether the train accelerates or decelerates by applyinga braking system
The event tree analysis for train collision is shown inFigure 4 Three barriersnamely Signal decelerate code bytrack circuit and Brake systems have been established todecrease the risk caused by the train collision Each of the bar-riers has two possible states ok or fail As a result of the analy-sis eight collision accidentsconsequences are distinguished
Table 4 Failure rate of components
Parameter Meaning Value120582InfSend Failure rate of code sending module 1119890 minus 7h120582InfRev Failure rate of code receiving module 1119890 minus 7h120582Monitor Failure rate of monitor 1119890 minus 5h120582CI Failure rate of CI 1119890 minus 6h120582ATP Failure rate of ATP 1119890 minus 06h120582Brake Failure rate of brake 7119890 minus 06h
Table 5 Corrective and proactive cost
Parameter TC Signal BrakSysRC119862 1500 500 3000RC119875 800 50 1000
Table 6 Product loss of different accident levels
Parameter 1198711
1198712
1198713
1198714
LC 5000 10000 30000 100000
For example when an external obstacle occupies the trackand the monitor system can successfully detect the presenceof the obstacle and send the information toCI via track circuit(TC) the CI will then inform the signal to display red aspect(ie signal is ok) At the same time the TC sends decelerationcode to train (ie TC is ok) and the braking system is normal(ie brake is ok) then the collision will be prevented and theconsequence is ldquosaferdquo On the other hand when an externalobstacle occupies the track and the monitor system TCCI signal and brake system all fail then the collision willbe inevitable and the resulting consequence is catastrophic(d7) Given the failure rate of the different components theldquoequivalent riskrdquo for each accident is estimated by the numer-ical results derived from the probability inference discussedin Section 4 above
Figure 5 illustrated the maintenance model of obsta-cle collision with high-speed train The model consists ofthree subsystems track circuit (TC) Signal and brake sys-tem (BrakSys) The reliability of subsystem depends on itsconstituted components For example the reliability of TCsubsystem depends on code sending module (InfSend) andcode receivingmodule (InfRev) signal subsystemonmonitorsystem (Monitor) and CI and brake subsystem on automatictrain protection (ATP) and brake equipment (Brake) Thefailure rate of components corrective and proactive cost andthe product loss of different accident levels are given in Tables4 5 and 6 respectively
The reliability probability distribution of componentSignal track circuit (TC) and Brake system (BrakSys) isshown in Figure 6 The total mission time is assumed to be31 time units (ie month) Thirty-one months are sufficientfor this purpose because for predictive maintenance it isinaccurate and meaningless to predict future deteriorationfor complex industrial system due to operational regulationenvironmental changes and human activityThe result of themean values of expected repair cost rate of Signal TC andBrake component is shown in Figure 7 Figure 8 illustrates the
8 Mathematical Problems in Engineering
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
InfSend InfRev Monitor CI ATP Brake
TC Signal BrakSys
Accident
(t minus 1) (t minus 1)
(t minus 1) (t minus 1)
(t minus 1)
(t minus 1)
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
(t minus 1) (t minus 1) (t minus 1) (t minus 1) (t) (t) (t) (t) (t) (t)
(t)(t)(t)
(t)
t slicet minus 1 slice
RC(t minus 1) RC(t)
Figure 5 Maintenance model for high- speed train
0 5 10 15 20 25 30 3505
06
07
08
09
1
11
12
Time unit (month)
Relia
bilit
y
SignalTCBrake
Figure 6 Reliability probability distribution of Signal TC andBrake component
mean values of total repair cost rate total production loss rateand the total cost rateThe latter is the sum of total repair costrate and total production loss rate From Figure 8 it can beseen that the optimalmaintenance time is 10 time units From(11) the corresponding reliability of for Signal TC and Brakecomponent is 098711 099856 and 094971 respectively
6 Conclusions
The paper presents a methodology for the optimization ofmaintenance strategies This approach ensures that not onlythe safety of equipment is increased but also that the cost
0 5 10 15 20 25 30 350
100
200
300
400
500
600
700
800
900
1000Ex
pect
ed re
pair
cost
rate
SignalTCBrake
Time unit (month)
Figure 7Mean values of expected repair cost rate of Signal TC andBrake component
of maintenance including the cost of failure is reduced Thework reportedwill contribute to the ldquoavailabilityrdquo of the safetycritical systems In order to calculate the failure probabilityand consequence of each failure scenario a maintenancemodel based on 2-TBN has been created An ad hoc inferenceprocedure along with its proof of correctness is provided toefficiently compute the probability of component failure ratesThe consequence of different failure scenarios is coded inconditional probability table (CPT) as part of the associatedmaintenance model In the approach proposed in the paperonly the systemrsquos optimal maintenance time was considered
Mathematical Problems in Engineering 9
0 5 10 15 20 25 30 350
200
400
600
800
1000
1200
1400
1600
1800
2000
Mai
nten
ance
cost
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
Total repair cost rateTotal product loss rateTotal cost rate
Time unit (month)
Figure 8 Mean values of total repair cost rate total production lossrate and the total cost rate
However the study can be extended so that each componentrsquosoptimal maintenance time can be calculated in the same way
Acknowledgments
The authors would like to thank the support of the Inter-national Science amp Technology Cooperation Program ofChina under Grant no 2012DFG81600 the Railway MinistryScience andTechnologyResearch andDevelopment Program(no 2013X015-B) and the State Key laboratory of Rail TrafficControl and Safety of Beijing Jiaotong University within theframe of the project (no RCS2012ZT005)
References
[1] J Lee J Ni D Djurdjanovic H Qiu and H Liao ldquoIntelligentprognostics tools and e-maintenancerdquo Computers in Industryvol 57 no 6 pp 476ndash489 2006
[2] J Hu L Zhang andW Liang ldquoOpportunistic predictive main-tenance for complex multi-component systems based on DBN-HAZOP modelrdquo Process Safety and Environmental Protectionvol 90 pp 376ndash388 2012
[3] L Krishnasamy F Khan and M Haddara ldquoDevelopment of arisk-basedmaintenance (RBM) strategy for a power-generatingplantrdquo Journal of Loss Prevention in the Process Industries vol18 no 2 pp 69ndash81 2005
[4] N S Arunraj and JMaiti ldquoRisk-basedmaintenance-techniquesand applicationsrdquo Journal of Hazardous Materials vol 142 no3 pp 653ndash661 2007
[5] F V Jensen S L Lauritzen and K G Olesen ldquoBayesian updat-ing in causal probablisitic networks by local computationsrdquoComputational Statistics Quarterly vol 4 pp 269ndash292 1990
[6] A P Dawid ldquoApplications of a general propagation algorithmfor probabilistic expert systemsrdquo Statistics and Computing vol2 no 1 pp 25ndash36 1992
[7] D Draper ldquoClustering without (thinking about) triangulationrdquoin Proceedings of the 11th Conference on Uncertainty in ArtificialIntelligence 1995
[8] X An Y Xiang and N Cercone ldquoDynamic multiagentprobabilistic inferencerdquo International Journal of ApproximateReasoning vol 48 no 1 pp 185ndash213 2008
[9] R Donat P Leray L Bouillaut and P Aknin ldquoA dynamicBayesian network to represent discrete duration modelsrdquo Neu-rocomputing vol 73 no 4-6 pp 570ndash577 2010
[10] R Dechter ldquoBucket elimination a unifying framework forreasoningrdquo Artificial Intelligence vol 113 no 1 pp 41ndash85 1999
