Leoni, Leonardo; Cantini, Alessandra; Bahootoroody ...

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Leoni Leonardo Cantini Alessandra Bahootoroody Farshad Khalaj Saeed De CarloFilippo Abaei Mohammad Mahdi BahooToroody AhmadReliability Estimation under Scarcity of Data A Comparison of Three Approaches

Published inMathematical Problems in Engineering

DOI10115520215592325

Published 20032021

Document VersionPublishers PDF also known as Version of record

Published under the following licenseCC BY

Please cite the original versionLeoni L Cantini A Bahootoroody F Khalaj S De Carlo F Abaei M M amp BahooToroody A (2021)Reliability Estimation under Scarcity of Data A Comparison of Three Approaches Mathematical Problems inEngineering 2021 1-15 [5592325] httpsdoiorg10115520215592325

Research ArticleReliability Estimation under Scarcity of Data A Comparison ofThree Approaches

Leonardo Leoni 1 Alessandra Cantini1 Farshad BahooToroody2 Saeed Khalaj2

Filippo De Carlo 1 Mohammad Mahdi Abaei3 and Ahmad BahooToroody4

1Department of Industrial Engineering (DIEF) University of Florence Florence 50123 Italy2Department of Civil Engineering University of Parsian Qazvin 3176795591 Iran3Department of Maritime and Transport Technology Delft University of Technology Delft 2628 CD Netherlands4Marine and Arctic Technology Group Department of Mechanical Engineering Aalto University Espoo 11000 Finland

Correspondence should be addressed to Filippo De Carlo filippodecarlounifiit

Received 13 January 2021 Revised 27 February 2021 Accepted 11 March 2021 Published 20 March 2021

Academic Editor Vahid Kayvanfar

Copyright copy 2021 Leonardo Leoni et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

During the last decades the optimization of the maintenance plan in process plants has lured the attention of many researchersdue to its vital role in assuring the safety of operations Within the process of scheduling maintenance activities one of the mostsignificant challenges is estimating the reliability of the involved systems especially in case of data scarcity Overestimating theaverage time between two consecutive failures of an individual component could compromise safety while an underestimate leadsto an increase of operational costs +us a reliable tool able to determine the parameters of failure modelling with high accuracywhen few data are available would be welcome For this purpose this paper aims at comparing the implementation of threepractical estimation frameworks in case of sparse data to point out the most efficient approach Hierarchical Bayesian modelling(HBM) maximum likelihood estimation (MLE) and least square estimation (LSE) are applied on data generated by a simulatedstochastic process of a natural gas regulating and metering station (NGRMS) which was adopted as a case of study +e resultsidentify the Bayesian methodology as the most accurate for predicting the failure rate of the considered devices especially for theequipment characterized by less data available +e outcomes of this research will assist maintenance engineers and assetmanagers in choosing the optimal approach to conduct reliability analysis either when sufficient data or limited data are observed

1 Introduction

Several hazardous substances are handled inside process plantstherefore unforeseen events could produce fires explosionsand chemical releases that could generate enormous financialloss and injuries or deaths of nearby employees and civilians[1] Among the potential causes asset failure is often regardedas the primary source of the aforementioned dangerousphenomena [2] hence the equipment involved in processindustries should be adequately maintained to guarantee ap-propriate standards of safety and reliability while generating aprofit from the operations

Over the past decades safety and reliability requirementshave progressively increased [3] leading to significant de-ployment of resources in maintenance activities [4] +isfundamental vision has resulted in the development of manymaintenance policies such as reliability-centered mainte-nance (RCM) [5ndash11] risk-based maintenance (RBM)[12ndash16] and condition-based maintenance (CBM) [8ndash11]Within the development of a CBM plan failure prognosis isof prominent importance thus there is an ongoing effort oncondition monitoring and calculation of the remaininguseful life [17ndash19] Zeng and Zio [18] developed a dynamicrisk assessment framework based on a Bayesian model to

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5592325 15 pageshttpsdoiorg10115520215592325

update the reliability of safety barriers as soon as new dataare collected After updating the reliabilities the risk indexesare determined through an event-tree (ET) Another rele-vant work by Chen et al [19] proposes an integration be-tween neuro-fuzzy systems (NFSs) and Bayesian algorithmto predict the evolution of the operating condition of a givensystem +e approach is tested on two case studies and theresults reveal a greater accuracy than other conventionalpredictors (eg recurrent neural networks NFSs and re-current NFSs)

To implement a proper maintenance plan based onpreventive actions a crucial task is represented by the es-timation of the probabilities of failure During this phase theaccuracy of the prediction is essential since overestimatingthe failure rates could lead to greater resource consumptionBy contrast an underestimation may delay the maintenanceactions resulting in a riskier state of the operations Con-sequently a great deal of research studies has been made toprovide accurate estimation procedures adopting differenttools such as fault tree analysis (FTA) [20] probability graphpaper [21] support vector machines [22] FORM (first-orderreliability method) [23] SORM (second-order reliabilitymethod) [24] and Bayesian network (BN) [25] To prove theadvantages and limitations of the estimation methodologiesmany researchers have also focused their efforts on com-paring the results arising from the application of distinctapproaches [26ndash28] Musleh and Helu [27] applied Bayesianinference MLE and LSE to censored data samples +eresults of this study pointed out the Bayesian estimator as thebest in terms of bias mean squared error and Pitmannearness probability A more recent work by BahooToroodyet al [28] presented the comparison between the MLE andthe HBM in case of perfect repair and minimal repair +eauthors tested the two approaches on an NGRMS provingthat the Bayesian inference provides more precision infailure modelling than the MLE +e proposed frameworksoperate under condition of availability of sufficient datawhile the challenges arising from sparse and limited datahave not been addressed

Within probability and reliability applications datascarcity is regarded as one of the main issues Indeed lack ofavailable data increases the uncertainties related to the es-timation process [29] causing sometimes the inability tofind the probability distributions [30] Sparse data could begenerated from many sources such as the rarity of an eventlimited knowledge missing data and impropriate datacollection Moreover the implementation of maintenancestrategies also contributes reducing available data sincemaintenance actions are performed to prevent failures fromhappening [31] Quite recently a BN-based quantitative riskassessment methodology was developed by Yang et al [32]In this work a BN along with precursors are adopted to copewith data scarcity while the consequences are evaluatedthrough loss functions +e classic Bayesian approaches canpartially compensate for limited data by incorporating priorknowledge and expert judgments However under theprimary assumption of the work they neglected the effect ofsource-to-source variability of failure data in the process

model [33] To overcome this limitation and simultaneouslydeal with sparse data HBM along with precursor data hasbeen extensively exploited by many academics [34ndash38] Liet al [39] integrated the BN and the HBM for a dynamic riskassessment of a submerged pipeline In their study theclassical BN is used to model the conditional dependenciesamong primary events while the hierarchical approach isdeveloped for predicting the probabilities associated withbasic events and the safety barriers +e proposed meth-odology can be updated as soon as new information becomesavailable by including them into the prior distributioncharacterizing the HBM

During the last years the adoption of HBM has spread toa broader audience thanks to the advances in open-sourceMarkov chain Monte Carlo (MCMC) sampling softwaresuch as OpenBugs [40] Examples of applications includeRBM planning [41 42] condition monitoring [43 44] andprobabilistic risk assessment [45 46] Recently Abaei et al[47] presented an HBM-based methodology able to predictthe probability of failure of a tidal energy converter assuminga homogeneous Poisson process (HPP) for the failuremodelling

Despite all the ongoing efforts there is still a need for asound tool able to deal with the uncertainties arising fromthe lack of data in maintenance applications Indeed whilethe literature provides many comparison studies amongdistinct statistical methodologies when sufficient data areobserved less interest has been devoted to the comparison ofestimation tools under the assumption of limited dataavailable To this end the main objective of this paper is toprovide a comparison between the Bayesian inference andtwo classic estimation approaches (ie MLE and LSE) in theevent of data scarcity arising from frequent preventivemaintenance actions +e methods are evaluated based ontheir accuracy in the estimation process for the failure rate ofthe components belonging to an NGRMS which is chosen asa case study

11 Hierarchical Bayesian Modelling +e first step requiredto conduct a statistical inference is collecting ldquoDatardquo whichare defined as the observed values of a given process NextldquoInformationrdquo is obtained by manipulating evaluating andorganizing ldquoDatardquo +e process of gathering ldquoInformationrdquoleads to acquire ldquoKnowledgerdquo which is subsequentlyexploited to perform ldquoInferencerdquo [48] As stated by El-Gheriani et al [29] the HBM allows to carry out the in-ference tasks through Bayesrsquo theorem shown by

π1(θ|x) f(x|θ)π0(θ)

1113938θf(x|θ)π0(θ)dθ (1)

Bayesrsquo theorem relies on the proportionality between theposterior distribution denoted by π1(θ|x) and the productof the likelihood function and the prior distribution re-spectively identified by f(x|θ) and π0(θ) +e prior dis-tribution is usually named informative when it concealsrelevant information about the unknown parameter of in-terest (θ) while it is regarded as noninformative when little

2 Mathematical Problems in Engineering

or no information about θ is considered [49] It is worthmentioning that the HBM owes its name to the adoption of amultistage or hierarchical prior distributions [50] given by[51]

π0(θ) 1113946emptyπ1(θ|φ)π2(φ)dφ (2)

where φ is a vector whose components are called hyper-parameters while π2(φ) is the hyperprior distributionsrepresenting the uncertainty of φ Finally π1(θ|φ) is referredas first-stage prior distribution which considers the vari-ability of θ given a certain value of φ

12 Maximum Likelihood Estimation Given a randomsample y (y1 y2 yn) arising from a stochastic pro-cess the objective of MLE is to determine the probabilitydistribution from which the sample is most likely to havebeen generated For this purpose it is required to specify aproper distribution for the sample data and its character-izing parameters Assuming that θ (θ1 θ2 θn) is avector that lies within the parameter space the mostprobable parameters that define the probability distributionof the observed data are obtained by maximising the like-lihood function illustrated by [28]

f θ1 θ2 θn|y( 1113857 f1 θ1|y( 1113857f2 θ2|y( 1113857 fn θn|y( 1113857

(3)

13 Least Square Estimation As stated by Myung [52] theLSE method is exploited primarily for descriptive purposesIts main goal is to define the parameters that generate themost accurate description of the observed data Lety (y1 y2 yn) be a sample of n observations and θ

(θ1 θ2 θm) a vector of parameters After choosing aproper distribution for the model the parameters that best-fit the data are found byminimizing the sum of squares error(SSE) shown by

SSE(θ) 1113944n

i1yi minus prdi(θ)( 1113857

2 (4)

where yi is the ith observation while prdi(θ) is the pre-diction of the model associated to the ith observation

+e remainder of the paper is organized as followsSection 2 describes the steps of the proposed study Section 3illustrates the implementation of the methodologies to theNGRMS while Section 4 provides the discussion of theresults At last in Section 5 conclusions are presented

2 Methodology

Within the reliability analysis process the exploitation ofdifferent estimation tools could lead to distinct resultswhich may affect the adopted maintenance strategy To thisend the main goal of this paper is to investigate the ap-plication of three estimation methodologies in case of fewdata available focusing mainly on the comparison betweenthe Bayesian approach and the classic approaches (ie MLE

and LSE) A brief overview of the framework is representedin Figure 1

+e first step (1) of the methodology is to collect failuredata generated by the considered process Since the majorityof industrial equipment undergoes substantial preventivemaintenance both Times To Failure (TTFs) and CensoredTimes To Failure (CTTFs) are taken into account for thestudy (11) moreover the number of failures observedduring a specified time span is also considered (12) CTTFsarise when preventive maintenance is performed or a givencomponent survives longer than the exposure time

During the second phase (2) the failure model isspecified In the present work the HPP is adopted formodelling the failure behaviour of the considered devices+e HPP describes a scenario where the interarrival timesbetween failures are independent and identically distributedaccording to an exponential distribution Due to the fre-quent preventive measures that completely restore the life ofthe active components the assumption of constant failurerate (ie number of failures independent upon a time) isregarded as appropriate for this study

Next the third part (3) consists of selecting the desiredestimation tool which is used to compute the failure rate ofeach apparatus (4) +ree estimation methodologies areevaluated in this paper (i) HBM (31) (ii) MLE (32) and(iii) LSE (33) +e results arising from the different methodsare then compared (5) to point out the most accurate andprecise estimator A special focus on the comparison be-tween the more recent Bayesian inference and the classicapproaches is presented

21 Hierarchical Bayesian Modelling Assuming an HPP forthe failure events of a given system the number of failures xexperienced during a timeframe equal to t can be obtainedvia

f(x|λ) (λt)

xe

minus λt

x x 0 1 (5)

where λ is the intensity characterizing the Poisson distri-bution ie the unknown parameter of interest As suggestedby Siu and Kelly [49] the first-stage prior representing thevariability of λ among different sources should be a betadistribution given by

π1(λ|α β) βαλαminus 1

eminus βλ

Γ(α) (6)

where α and β are the hyperparameters which are con-sidered as independent before including any observationsinto the analysis [48] After choosing the likelihood and theprior distributions theMCMC simulations are performed todetermine the posterior distributions of the hyper-parameters As a result the posterior distribution of λ is alsoobtained through the sampling procedure

22 Maximum Likelihood Estimation Under the hypothesisof HPP the probability distribution of failure interarrivaltimes denoted by T is given by the following equations

Mathematical Problems in Engineering 3

F(t) 1 minus eminus λt

(7)

f(t) λeminus λt

(8)

where λ represents the rate of arrival As previously dis-cussed the MLE determines the parameters of the consid-ered distribution by maximising the likelihood functionwhich in case of exponential distribution is expressed by

L(λ) 1113945n

i1λe

minus λTi λne

minus λ1113936n

i1 Ti( 1113857 λn

e(minus λnT)

(9)

where n indicates the total number of failures while T isaddressed as the mean of the interarrival times +e esti-mator of λ is then found via [28 53]

1113954λ 1T

n

1113936ni1 Ti

(10)

23 Least Square Estimation Let the interarrival time offailures follow a negative exponential distribution describedby equations (7) and (8) +e LSE estimates the unknownparameters by defining a straight line that minimizes thesum of squared distances between the observed data and theline itself +erefore the exponential distribution should berewritten in the form shown by

y ax + b (11)

After applying the logarithm to both sides of equation (7)and some simplification the following equation is obtained

t minusln[1 minus F(t)]

λ (12)

which represents a straight line with a minus1λ and b 0while y t and x ln[F(t)] Let t (t1 t2 tn) be asample of TTFs and the estimation of λ is found by min-imizing the SSE reported by

SSE(λ) 1113944n

i1ti minus

minusln 1 minus F ti( 11138571113858 1113859

λ1113888 1113889

2

(13)

where ti stands for the ith observed TTF while F(ti) isreplaced by the median rank expressed by [54]

F ti( 1113857 i minus 03n + 04

ti i 1 2 n t1 lt t2 middot middot middot lt tn( 1113857 (14)

3 Application of the Methodology to NGRMS

To show a practical application of the three approaches andcompare their results an NGRMS (Figure 2) is chosen as acase study A generic NGRMS is divided into four groupsand twelve main components listed in Table 1

+e natural gas distribution network is a complex in-frastructure formed by pipes and apparatuses able towithstand high-pressure values+us before distributing themethane to the final users the gas pressure must be reducedto be suitable for the various utilities To fulfill this taskNGRMSs are usually installed along with additional sub-sequent pressure reduction units +e core of the plant is thereduction group in which the pressure regulator and thepilot are tasked with reducing the pressure During standard

1 Data collection

2 Develop a failure model

3 Selection of estimation tool

32 Maximum likelihood estimation

33 Least square estimation

31 Hierarchical bayesian modelling

11 Collect times to failure

12 Collect number of failures

4 Estimate unknown parameters of interest

5 Result comparison

Figure 1 Flowchart representing the steps of the presented study


conditions the pressure is decreased by varying the cross-sectional flow area of the pressure regulator while thedownstream pilot is activated just in case a faster or moreaccurate pressure reduction is required +e solid and liquidimpurities that could be present in the gas flow are removedby the filter which is located upstream of the pressureregulator Since decreasing the pressure is always accom-panied by a temperature reduction the gas must be heatedbefore entering the pressure regulator to avoid the formationof ice To this end a water flow is preheated by a boiler andsubsequently it is sent to an exchanger in which flows themethane gas At last the measuring group evaluates thenatural gasrsquos most relevant parameters (eg pressuretemperature and mass flow) while the odorization group isrequired to add a precise quantity of odorizer usually tet-rahydrothiophene (THT) to the gas flow

31 Data Collection To implement the approaches theoperations of a real-life NGRMS were reproduced throughthe AnyLogic simulation software (developed by +e

AnyLogic Company httpwwwxjtekcom) focusing on thestochastic failure generation process +e developed modelhas an NGRMS located in Tuscany near Arezzo Town and amaintenance centre in Prato Town (Figure 3) served by twomaintenance teams available 247 +e first maintenancesquad is tasked with preventive actions while the second oneis in charge of the corrective actions Agent-based modellingand fifteen simulation run were adopted for this study Fromeach run the TTFs the CTTFs and the number of failureswere extracted to conduct the subsequent analysis (step 1 ofFigure 1) It is worthwhile mentioning that the failure ratesadopted for the simulation are chosen based on real expe-rience therefore from now on they are considered as ldquorealrdquoparameters As a result such failure rates are also exploitedas reference values to compare the precision of the threeestimation approaches

For the sake of conciseness the estimation methods willbe presented in detail for the pressure regulator however asummary reporting the obtained results for all the consid-ered components will be discussed later through this studyTable 2 shows the observed number of failures and the

Table 1 NGRMSrsquo main groups and components

Group ComponentReduction Pressure regulator (PR)

PilotFilter

Measuring Pressure and temperature gauge (PTG)CalculatorMeter

Remote control system (RCS)Odorization THT tank

THT pipelinesPreheating Pump

BoilerWater pipe (WP)

Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O

Component1 Regulation group2 Measuring group3 Odorization group4 Boiler5 Pump6 Valve 7 Filter8 Exchanger9 Pressure regulator10 Pilot

8 9

Figure 2 Schematic representation of an NGRMS


number of preventivemaintenance actions arising from eachsimulation run for the pressure regulator Each simulationrun is considered as a different source of the failure rate ofthe components for the HBM After this brief introductionstep 4 of Figure 1 will be implemented in the next sections

32 Hierarchical Bayesian Modelling +e BN illustrated byFigure 4 was adopted to predict the posterior distribution ofλ +e aforementioned number of failures observed in eachsource (Table 2) is denoted by Xi while λi refers to the failurerate of the i-th run +e calculation of posterior distributionin Bayesian will be carried out by MCMC simulation +reechains each starting from a distinct point in the parameterspace were used to assure the convergence +e samplingfrom the likelihood and the prior distribution was conductedwith 105 iterations for each chain preceded by 1000 burn-initerations +e estimated posterior distribution of α and βalong with their respective mean values is shown in Figure 5Furthermore the correlation between the two parameters isrepresented in Figure 6

+eMCMC sampling process revealed a mean value of αequal to 03863 with a 95 credible interval of (0115609065) while the mean of the posterior predicted distri-bution for β is 168E+ 4 hours with the following 95credible interval (212E+ 3 452E+ 4)

+e caterpillar plot representing the 95 credible in-terval for the failure rate of each source is illustrated inFigure 7 As shown in Figure 7 the computed mean value(red vertical line) of the posterior predictive distribution forλ is 168Eminus 05 (per hour) Table 3 reports a summary of theposterior distribution of every λi

Due to the different number of failures and distinctexposure time observed in each source the mean failurerate varies significantly from source-to-source For in-stance the first source is characterized by the highest meanfailure rate of 569E-05 (per hour) since it has experiencedthe highest number of failures (3) in the shortest timespan(44300 hours) By contrast the tenth pressure regulatorowns the lowest mean failure rate equal to 359E06 (perhour) because no failure has been detected for a decade+e source-to-source variability is incorporated within theaforementioned mean value of 168Eminus 05 (per hour)

Considering an exponential distribution for the inter-arrival time of failures the MTTF is given by the reciprocalof the failure rate as shown in

MTTF 1λ (15)

Following equation (15) the MTTF of the pressureregulator is estimated It emerged that the average timebetween two subsequent failure states is 59524 hours (aboutsix and a half years)

33 Maximum Likelihood Estimation To perform this stepof the analysis the statistical software called Minitab wasexploited Minitab allows considering both the TTFs and theCTTFs for the estimation of λ +e MLE application pro-vides a failure rate of 122Eminus 05 (per hour) which

corresponds to an MTTF equal to 82060 hours (more thannine years) +e resulting probability density function isreported in Figure 8

34 Least Square Estimation As in the previous stepMinitab was adopted for the LSE as well +e intensity of thePoisson process estimated by the LSE method is slightlyhigher than the rate calculated via the MLE +e calculationdepicted a λ equal to 128Eminus 05(per hour) equivalent to anMTTF of 78219 hours (slightly less than nine years) +eexponential probability density function of failure inter-arrival time corresponding to the estimated failure rate isillustrated in Figure 9

4 Discussion

In this section step 5 of Figure 1 is presented As describedby the previous section applying the three approaches withthe same input data provided three different values for thefailure rate of the pressure regulator +e HBM yields afailure rate of λHBM 168Eminus 05 which results in an MTTFequal to 59524 hours On the other hand it emerged that theMTTFs calculated by the MLE and LSE approaches aremuch higher than the Bayesian ones Indeed the MLE andthe LSE quantified an average time between two subsequentfailures of 82060 and 78219 hours respectively

+e real failure rate (ie the one adopted during thesimulation process) is λREAL 164Eminus 05 corresponds to aMTTF of 60882 hours Accordingly the calculation revealeda much accurate and precise Bayesian estimator with respectto the other ones Indeed the real value is underestimated bythe HBM for 1300 hours (about 54 days) while the otherestimations are more than 2 years longer compared toMTTFREAL

A summary of the Bayesian results and the other ap-proaches is listed in Tables 4 and 5 +e comparison will bediscussed further for each group in the following sections

+e estimated MTTFs are transformed into dimen-sionless values through the average time between twoconsecutive failures adopted for the simulation (ieMTTFREAL) +e results are shown in Table 6 and Figure 10

41 Reduction Group To illustrate the differences arisingfrom the three estimation methodologies the cumulativedistribution functions (CDFs) of each approach were de-veloped for the reduction devices (Figures 11 and 12) Asdepicted by Tables 4ndash6 the HBM provided the most accurateestimation for the failure rate of the pilot while the MLE andthe LSE of the MTTF are respectively 129 days longer and154 days shorter than the real value Considering the filterthe difference between MTTFREAL and MTTFHBM is just17 hours while both the MLE and the LSE overestimate theaverage time between two consecutive failures +e MLEyields an MTTF which is 20 days longer compared toMTTFREAL whereas the mean time span between two failurestates is estimated by the LSE as 25 days longer than the realone


42 Measuring Group For the calculator the HBM yields aposterior mean interarrival time of failure equal to 68027hours while the MLE and the LSE of MTTF are estimated at88825 and 80837 hours respectively Given the real value of73233 hours the HBM is the most accurate estimation toolonce again +e Bayesian approach manifested its advan-tages over the other methodologies for the PTG as wellIndeed the difference between MTTFHBM and MTTFREAL is

128 hours (about five days) while both the MLE and the LSEoverestimate the real mean time between two contiguousfailures by approximately 1000 hours (41 days) On thecontrary the application of LSE emerged as the most precisefor both the meter and the RCS However the Bayesianinference demonstrated greater accuracy than the MLE forthese two devices +eMTTFHBM of the meter is just 14 dayslonger than the MTTF estimated by the LSE while the

α β

λ1 λ2 λn

X1 X2 Xn

Figure 4 Developed HBM for estimating the failure rate of each device

Figure 3 Map of the location of the simulated case

Table 2 Number of failures preventive actions and exposure time for the pressure regulator in each source (simulation run)

Source Number of failures Number of preventive actions Exposure time (hours)1 3 0 443002 1 0 443003 2 0 443004 0 1 788405 0 1 788406 0 1 788407 0 0 540008 1 0 540009 0 0 5400010 0 1 8760011 2 0 8760012 2 0 8760013 1 0 6132014 1 0 6132015 0 0 61320


discrepancy between the Bayesian prediction and the LSEestimator for the RCS is equal to 12 days Both the HBM andthe LSE showed an estimation error of about 5000 hours

(208 days) and 1500 hours (62 days) for the meter and theRCS respectively +e CDFs of the measuring componentsare represented in Figures 13 and 14

25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)

Figure 5 +e posterior probability density function for alpha (on the left) and beta (on the right)

00 05 10 15Alpha

70000

50000

30000

100000

Beta

Figure 6 Correlation between alpha and beta

[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5

Caterpillar plot lambda

00 50E ndash 5 10E ndash 4Lambda

Figure 7 Predicted 95 credible interval for the failure rate of the pressure regulator in each source +e black dots are the posterior meansof each source while the red line represents the average of posterior means


Table 3 Statistical properties of the failure rate for each source of the pressure regulator

HBM parameter Mean 25 percentile 975 percentileλ1 569Eminus 05 129Eminus 05 136Eminus 04λ2 230Eminus 05 129Eminus 06 753Eminus 05λ3 398Eminus 05 607Eminus 06 107Eminus 04λ4 391Eminus 06 157Eminus 12 233Eminus 05λ5 392Eminus 06 161Eminus 12 230Eminus 05λ6 395Eminus 06 180Eminus 12 232Eminus 05λ7 528Eminus 06 206Eminus 12 310Eminus 05λ8 197Eminus 05 110Eminus 06 640Eminus 05λ9 527Eminus 06 260Eminus 12 309Eminus 05λ10 359Eminus 06 156Eminus 12 212Eminus 05λ11 229Eminus 05 349Eminus 06 605Eminus 05λ12 230Eminus 05 350Eminus 06 604Eminus 05λ13 178Eminus 05 991Eminus 07 582Eminus 05λ14 178Eminus 05 978Eminus 07 582Eminus 05λ15 477Eminus 06 216Eminus 12 279Eminus 05

0 50000 100000 150000 200000Interarrival time of failure

10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h

Figure 8 Interarrival time of failure distribution for the pressure regulator obtained via MLE


10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h

Figure 9 Interarrival time of failure distribution for the pressure regulator obtained via LSE

Table 4 Real failure rate and failure rates estimated through the three approaches

Component Real λ λ HBM λ MLE λ LSEPressure regulator 164Eminus 05 168Eminus 05 122Eminus 05 128Eminus 05Pilot 218Eminus 05 218Eminus 05 204Eminus 05 237Eminus 05Filter 967Eminus 05 969Eminus 05 926Eminus 05 915Eminus 05PTG 517Eminus 05 520Eminus 05 491Eminus 05 496Eminus 05Calculator 137Eminus 05 147Eminus 05 112Eminus 05 124Eminus 05Meter 226Eminus 05 201Eminus 05 184Eminus 05 202Eminus 05RCS 267Eminus 05 255Eminus 05 254Eminus 05 257Eminus 05THT tank 108Eminus 05 102Eminus 05 818Eminus 06 942Eminus 06THT pipeline 755Eminus 06 779Eminus 06 617Eminus 06 976Eminus 06Pump 480Eminus 05 482Eminus 05 450Eminus 05 431Eminus 05Boiler 316Eminus 05 324Eminus 05 319Eminus 05 325Eminus 05Water pipe 740Eminus 06 549Eminus 06 411Eminus 06 448Eminus 06


43 Odorization Group +e CDFs built for the THT tankand THT pipe are illustrated in Figure 15 +e Bayesianapproach proved its higher performance also for the com-ponents belonging to the odorization group +e MTTFHBM

of the THT tank is estimated at 98039 hours which is about5500 hours (about 230 days) shorter than MTTFREAL +eMLE model resulted in an average interarrival time of 139years while the LSE yields anMTTF of 121 years Compared

Table 5 Real MTTF and MTTFs estimated through the three approaches

Component Real MTTF MTTF HBM MTTF MLE MTTF LSEPressure regulator 60882 59524 82060 78219Pilot 45815 45872 48909 42123Filter 10337 10320 10800 10934PTG 19359 19231 20379 20151Calculator 73233 68027 88825 80837Meter 44248 49751 54343 49400RCS 37492 39216 39392 38928THT tank 92593 98039 122272 106137THT pipeline 132363 128370 162050 102454Pump 20848 20747 22231 23219Boiler 31623 30864 31362 30741Water pipe 135166 182149 243075 223270

Table 6 Dimensionless mean time to failure (D-MTTF) for each estimation approach

Component D-MTTF HBM D-MTTF MLE D-MTTF LSEPressure regulator 0978 1348 1285Pilot 1001 1068 0919Filter 0998 1045 1058PTG 0993 1053 1041Calculator 0929 1213 1104Meter 1124 1228 1116RCS 1046 1051 1038THT tank 1059 1321 1146THT pipeline 0970 1224 0774Pump 0995 1066 1114Boiler 0976 0992 0972Water pipe 1348 1798 1652

18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE

Figure 10 Developed dotplot of the D-MTTFs for each methodology +e black dashed line represents the real MTTF


to the real value the ML and the LSE estimators showed agap of about 30000 hours (more than three years) and13000 hours (about 15 years) respectively For the THTpipe a similar scenario is seen Indeed both the ML and LSof MTTFs are about 30000 hours longer than the real av-erage time expected before experiencing a failure By con-trast the HBM predicted a posterior mean value of 128370hours which is close to the real mean interarrival time tofailure of 132363 hours

44 Preheating Group +e water pipe is the componentassociated with the worst estimations due to extreme datascarcity (Figures 16 and 17) +e HBM yields a posteriorMTTF of 182149 hours which is five years longer thanMTTFREAL +e MLE and the LSE also overestimated thereal average time between two consecutive failures by 12 and10 years respectively Considering the boiler the MLE es-timator is the most accurate with a discrepancy of just 300hours (almost 13 days) compared to MTTFREAL Never-theless the application of the HBM is more precise than LSEAt last the Bayesian inference emerged as the best estimator

for the pump presenting a gap of 100 hours (four days) withrespect to the mean interarrival time of failure adopted forthe simulation On the other side an overestimation of 58and 99 days is observed respectively for the MLE and theLSE of the MTTF related to the pump

45 Discussion Maintenance Application +e HBM hasproven itself as the most reliable estimator under limiteddata which concerns particularly the pressure regulator thecalculator the THT tank the THTpipe and the water pipeIndeed these components are characterized by a longerMTTF than the other apparatuses therefore fewer failuresare observed during the same time interval +e Bayesianpredictions of the failure parameter for the aforementioneddevices show a better precision than the other estimationmethodologies Moreover the accuracy showed by the HBMis also higher than the other approaches for most of thedevices+e root mean square error (RMSE) is calculated foreachmethod to demonstrate the last statements as shown byequation (18)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)

Figure 11 Developed CDFs for the pressure regulator and the filter

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06

Figure 12 Developed CDF for the pilot


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)

Figure 14 Developed CDFs for the PTG and the RCS

RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)

Figure 15 Developed CDFs for the THT tank and the THT pipe

RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)

Figure 13 Developed CDFs for the calculator and the meter


RMSE

1113936ni1 MTTFi minus MTTFREAL i1113872 1113873

2

n

1113971

(16)

where n denotes the number of components while MTTFi isthe estimated average time between two consecutive failuresfor the ith device At last MTTFREAL i is the mean inter-arrival time between failures adopted for the ith equipmentduring the simulation +e RMSE computed for the HBMthrough equation (16) is equal to 13892 hours while theRMSE of the MLE and LSE is estimated respectively at34420 and 27761 hours Accordingly the exploitation of theBayesian method will result in a much safer maintenancestrategy without overlooking economic aspects by avoidingpremature maintenance actions

5 Conclusions

Any maintenance policy is deeply affected by the previousfailure rate estimation process which often suffers fromlimited data +us one of the most significant challengesassociated with the reliability analysis is selecting a properestimation approach capable of producing accurate and

precise results in case of limited data To this end the ap-plication of three estimation tools is investigated in thispaper with a particular focus on the comparison betweenthe Bayesian inference and two common estimationmethodologies the MLE and the LSE +e three analyseswere tested on twelve components of an NGRMS whoseoperations were simulated through a simulation model toextract failure data (ie TTF CTTF and the number offailures) Under the assumption of HPP the results high-lighted a greater accuracy of the HBM which emerged as themost precise estimator for nine devices out of twelve +eadvantages of the Bayesian estimator are especially evidentin the event of data shortage associated with the devices withgreater MTTF Indeed the lack of data is partially com-pensated by the HBM through the consideration of source-to-source variability which is disregarded by the MLE andthe LSE On the other side the MLE and LSE precisionimproves for the equipment characterized by more dataavailable up to taking the upper hand over the Bayesianinference for the meter the RCS and the boiler Howeverthe discrepancy between the Bayesian predictions and theother estimations for these components are negligible sincealmost no difference can be seen from their respective CDFs

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)

Figure 16 Developed CDFs for the water pipe and the boiler

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08

Figure 17 Developed CDF for the pump


Considering all the above adopting a Bayesian approach forthe reliability analysis will help to deal with sparse dataresulting in a more efficient and effective maintenance planFurther developments can include the application of weakly-informative kind of prior to the Bayesian model to incor-porate some prior knowledge into the estimationframework

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] H-m Kwon ldquo+e effectiveness of process safety management(PSM) regulation for chemical industry in Koreardquo Journal ofLoss Prevention in the Process Industries vol 19 no 1pp 13ndash16 2006

[2] F Jaderi Z Z Ibrahim and M R Zahiri ldquoCriticality analysisof petrochemical assets using risk based maintenance and thefuzzy inference systemrdquo Process Safety and EnvironmentalProtection vol 121 pp 312ndash325 2019

[3] H Soltanali A Rohani M H Abbaspour-Fard A Paridaand J T Farinha ldquoDevelopment of a risk-based maintenancedecision making approach for automotive production linerdquoInternational Journal of System Assurance Engineering andManagement vol 11 no 1 pp 236ndash251 2020

[4] Y Wang G Cheng H Hu and W Wu ldquoDevelopment of arisk-based maintenance strategy using FMEA for a contin-uous catalytic reforming plantrdquo Journal of Loss Prevention inthe Process Industries vol 25 no 6 pp 958ndash965 2012

[5] D Li and J Gao ldquoStudy and application of reliability-centeredmaintenance considering radical maintenancerdquo Journal ofLoss Prevention in the Process Industries vol 23 no 5pp 622ndash629 2010

[6] H Islam ldquoReliability-centeredmaintenance methodology andapplication a case studyrdquo Engineering vol 2 no 11 2010

[7] B Yssaad and A Abene ldquoRational reliability centeredmaintenance optimization for power distribution systemsrdquoInternational Journal of Electrical Power amp Energy Systemsvol 73 pp 350ndash360 2015

[8] G Zou K Banisoleiman A Gonzalez and M H FaberldquoProbabilistic investigations into the value of information acomparison of condition-based and time-based maintenancestrategiesrdquo Ocean Engineering vol 188 Article ID 1061812019

[9] C Duan Z Li and F Liu ldquoCondition-based maintenance forship pumps subject to competing risks under stochasticmaintenance qualityrdquo Ocean Engineering vol 218 Article ID108180 2020

[10] Z Liang B Liu M Xie and A K Parlikad ldquoCondition-basedmaintenance for long-life assets with exposure to operationaland environmental risksrdquo International Journal of ProductionEconomics vol 221 Article ID 107482 2020

[11] J Kang Z Wang and C Guedes Soares ldquoCondition-basedmaintenance for offshore wind turbines based on supportvector machinerdquo Energies vol 13 no 14 p 3518 2020

[12] G Pui J Bhandari E Arzaghi R Abbassi and V GaraniyaldquoRisk-based maintenance of offshore managed pressuredrilling (MPD) operationrdquo Journal of Petroleum Science andEngineering vol 159 pp 513ndash521 2017

[13] H Hu G Cheng Y Li and Y Tang ldquoRisk-based maintenancestrategy and its applications in a petrochemical reformingreaction systemrdquo Journal of Loss Prevention in the ProcessIndustries vol 22 no 4 pp 392ndash397 2009

[14] M Florian and J D Soslashrensen ldquoRisk-based planning of op-eration and maintenance for offshore wind farmsrdquo EnergyProcedia vol 137 pp 261ndash272 2017

[15] G Kumar and J Maiti ldquoModeling risk based maintenanceusing fuzzy analytic network processrdquo Expert Systems withApplications vol 39 no 11 pp 9946ndash9954 2012

[16] A BahooToroody MM Abaei E Arzaghi F BahooToroodyF De Carlo and R Abbassi ldquoMulti-level optimization ofmaintenance plan for natural gas system exposed to deteri-oration processrdquo Journal of Hazardous Materials vol 362pp 412ndash423 2019

[17] D Zhang A D Bailey and D Djurdjanovic ldquoBayesianidentification of hidden Markov models and their use forcondition-based monitoringrdquo IEEE Transactions on Reli-ability vol 65 no 3 pp 1471ndash1482 2016

[18] Z Zeng and E Zio ldquoDynamic risk assessment based onstatistical failure data and condition-monitoring degradationdatardquo IEEE Transactions on Reliability vol 67 no 2pp 609ndash622 2018

[19] C Chen B Zhang and G Vachtsevanos ldquoPrediction ofmachine health condition using neuro-fuzzy and Bayesianalgorithmsrdquo IEEE Transactions on Instrumentation andMeasurement vol 61 no 2 pp 297ndash306 2011

[20] D Yuhua and Y Datao ldquoEstimation of failure probability ofoil and gas transmission pipelines by fuzzy fault tree analysisrdquoJournal of Loss Prevention in the Process Industries vol 18no 2 pp 83ndash88 2005

[21] T Zhang and R Dwight ldquoChoosing an optimal model forfailure data analysis by graphical approachrdquo Reliability En-gineering amp System Safety vol 115 pp 111ndash123 2013

[22] J-M Bourinet F Deheeger andM Lemaire ldquoAssessing smallfailure probabilities by combined subset simulation andsupport vector machinesrdquo Structural Safety vol 33 no 6pp 343ndash353 2011

[23] A P Teixeira C Guedes Soares T A Netto and S F EstefenldquoReliability of pipelines with corrosion defectsrdquo InternationalJournal of Pressure Vessels and Piping vol 85 no 4pp 228ndash237 2008

[24] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132no 10 2010

[25] J Wu R Zhou S Xu and Z Wu ldquoProbabilistic analysis ofnatural gas pipeline network accident based on Bayesiannetworkrdquo Journal of Loss Prevention in the Process Industriesvol 46 pp 126ndash136 2017

[26] R Calabria and G Pulcini ldquoOn the maximum likelihood andleast-squares estimation in the inverse Weibull distributionrdquoStatistica Applicata vol 2 no 1 pp 53ndash66 1990

[27] R M Musleh and A Helu ldquoEstimation of the inverse Weibulldistribution based on progressively censored data compar-ative studyrdquo Reliability Engineering amp System Safety vol 131pp 216ndash227 2014

[28] A BahooToroody M M Abaei E Arzaghi et al ldquoOn reli-ability challenges of repairable systems using hierarchicalbayesian inference and maximum likelihood estimationrdquo


Process Safety and Environmental Protection vol 135pp 157ndash165 2020

[29] M El-Gheriani F Khan D Chen and R Abbassi ldquoMajoraccident modelling using spare datardquo Process Safety andEnvironmental Protection vol 106 pp 52ndash59 2017

[30] N Khakzad ldquoSystem safety assessment under epistemicuncertainty using imprecise probabilities in Bayesian net-workrdquo Safety Science vol 116 pp 149ndash160 2019

[31] D F Percy ldquoMaintenance based on limited datardquo in ComplexSystemMaintenance Handbook pp 133ndash154 Springer BerlinGermany 2008

[32] M Yang F Khan and P Amyotte ldquoOperational risk as-sessment a case of the Bhopal disasterrdquo Process Safety andEnvironmental Protection vol 97 pp 70ndash79 2015

[33] P Kumari D Lee Q Wang M N Karim and J S-I KwonldquoRoot cause analysis of key process variable deviation for rareevents in the chemical process industryrdquo Industrial amp Engi-neering Chemistry Research vol 59 no 23 2020

[34] N Khakzad F Khan and N Paltrinieri ldquoOn the applicationof near accident data to risk analysis of major accidentsrdquoReliability Engineering amp System Safety vol 126 pp 116ndash1252014

[35] N Khakzad S Khakzad and F Khan ldquoProbabilistic riskassessment of major accidents application to offshoreblowouts in the Gulf of Mexicordquo Natural Hazards vol 74no 3 pp 1759ndash1771 2014

[36] M Yang F I Khan and L Lye ldquoPrecursor-based hierarchicalBayesian approach for rare event frequency estimation a caseof oil spill accidentsrdquo Process Safety and EnvironmentalProtection vol 91 no 5 pp 333ndash342 2013

[37] M Yang F Khan L Lye and P Amyotte ldquoRisk assessment ofrare eventsrdquo Process Safety and Environmental Protectionvol 98 pp 102ndash108 2015

[38] H Yu F Khan and B Veitch ldquoA flexible hierarchicalBayesian modeling technique for risk analysis of major ac-cidentsrdquo Risk Analysis vol 37 no 9 pp 1668ndash1682 2017

[39] X Li G Chen F Khan and C Xu ldquoDynamic risk assessmentof subsea pipelines leak using precursor datardquo Ocean Engi-neering vol 178 pp 156ndash169 2019

[40] D Spiegelhalter A+omas N Best andD LunnOpenBUGSUser Manual Version 30 2 MRC Biostatistics Unit Cam-bridge UK 2007

[41] L Leoni A BahooToroody M M Abaei et al ldquoOn hierar-chical bayesian based predictive maintenance of autonomousnatural gas regulating operationsrdquo Process Safety and Envi-ronmental Protection vol 147 pp 115ndash124 2021

[42] L Leoni F De Carlo F Sgarbossa and N PaltrinierildquoComparison of risk-based maintenance approaches appliedto a natural gas regulating and metering stationrdquo ChemicalEngineering Transactions vol 82 pp 115ndash120 2020

[43] A BahooToroody F De Carlo N Paltrinieri M Tucci andP H A J M Van Gelder ldquoBayesian regression based con-dition monitoring approach for effective reliability predictionof random processes in autonomous energy supply opera-tionrdquo Reliability Engineering amp System Safety vol 201 ArticleID 106966 2020

[44] A BahooToroody M M Abaei F BahooToroodyF De Carlo R Abbassi and S Khalaj ldquoA condition moni-toring based signal filtering approach for dynamic time de-pendent safety assessment of natural gas distribution processrdquoProcess Safety and Environmental Protection vol 123pp 335ndash343 2019

[45] L Niu P H A J M Van Gelder Y Guan C Zhang andJ K Vrijling ldquoProbabilistic analysis of phytoplankton

biomass at the Frisian Inlet (NL)rdquo Estuarine Coastal and ShelfScience vol 155 pp 29ndash37 2015

[46] M Yang F Khan V Garaniya and S Chai ldquoMultimedia fatemodeling of oil spills in ice-infested waters an exploration ofthe feasibility of fugacity-based approachrdquo Process Safety andEnvironmental Protection vol 93 pp 206ndash217 2015

[47] M M Abaei N R Arini P R +ies and J Lars ldquoFailureestimation of offshore renewable energy devices based onhierarchical bayesian approachrdquo in Proceedings of the Inter-national Conference on Offshore Mechanics and ArcticEngineering Prague Czech June 2019

[48] D L Kelly and C L Smith ldquoBayesian inference in proba-bilistic risk assessment-+e current state of the artrdquo ReliabilityEngineering amp System Safety vol 94 no 2 pp 628ndash643 2009

[49] N O Siu and D L Kelly ldquoBayesian parameter estimation inprobabilistic risk assessmentrdquo Reliability Engineering amp Sys-tem Safety vol 62 no 1-2 pp 89ndash116 1998

[50] D Kelly and C Smith Bayesian Inference for Probabilistic RiskAssessment A Practitionerrsquos Guidebook Springer Science ampBusiness Media Berlin Germany 2011

[51] M M Abaei E Arzaghi R Abbassi V GaraniyaM Javanmardi and S Chai ldquoDynamic reliability assessmentof ship grounding using Bayesian Inferencerdquo Ocean Engi-neering vol 159 pp 47ndash55 2018

[52] I J Myung ldquoTutorial on maximum likelihood estimationrdquoJournal of Mathematical Psychology vol 47 no 1 pp 90ndash1002003

[53] S M Ross Introduction to Probability Models AcademicPress Cambridge MA USA 2014

[54] M Z Rashid and A S Akhter ldquoEstimation accuracy of ex-ponential distribution parametersrdquo Pakistan Journal of Sta-tistics and Operation Research vol 7 no 2 2011


Research ArticleReliability Estimation under Scarcity of Data A Comparison ofThree Approaches

Leonardo Leoni 1 Alessandra Cantini1 Farshad BahooToroody2 Saeed Khalaj2

Filippo De Carlo 1 Mohammad Mahdi Abaei3 and Ahmad BahooToroody4

1Department of Industrial Engineering (DIEF) University of Florence Florence 50123 Italy2Department of Civil Engineering University of Parsian Qazvin 3176795591 Iran3Department of Maritime and Transport Technology Delft University of Technology Delft 2628 CD Netherlands4Marine and Arctic Technology Group Department of Mechanical Engineering Aalto University Espoo 11000 Finland

Correspondence should be addressed to Filippo De Carlo filippodecarlounifiit

Received 13 January 2021 Revised 27 February 2021 Accepted 11 March 2021 Published 20 March 2021

Academic Editor Vahid Kayvanfar

Copyright copy 2021 Leonardo Leoni et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

During the last decades the optimization of the maintenance plan in process plants has lured the attention of many researchersdue to its vital role in assuring the safety of operations Within the process of scheduling maintenance activities one of the mostsignificant challenges is estimating the reliability of the involved systems especially in case of data scarcity Overestimating theaverage time between two consecutive failures of an individual component could compromise safety while an underestimate leadsto an increase of operational costs +us a reliable tool able to determine the parameters of failure modelling with high accuracywhen few data are available would be welcome For this purpose this paper aims at comparing the implementation of threepractical estimation frameworks in case of sparse data to point out the most efficient approach Hierarchical Bayesian modelling(HBM) maximum likelihood estimation (MLE) and least square estimation (LSE) are applied on data generated by a simulatedstochastic process of a natural gas regulating and metering station (NGRMS) which was adopted as a case of study +e resultsidentify the Bayesian methodology as the most accurate for predicting the failure rate of the considered devices especially for theequipment characterized by less data available +e outcomes of this research will assist maintenance engineers and assetmanagers in choosing the optimal approach to conduct reliability analysis either when sufficient data or limited data are observed

1 Introduction

Several hazardous substances are handled inside process plantstherefore unforeseen events could produce fires explosionsand chemical releases that could generate enormous financialloss and injuries or deaths of nearby employees and civilians[1] Among the potential causes asset failure is often regardedas the primary source of the aforementioned dangerousphenomena [2] hence the equipment involved in processindustries should be adequately maintained to guarantee ap-propriate standards of safety and reliability while generating aprofit from the operations

Over the past decades safety and reliability requirementshave progressively increased [3] leading to significant de-ployment of resources in maintenance activities [4] +isfundamental vision has resulted in the development of manymaintenance policies such as reliability-centered mainte-nance (RCM) [5ndash11] risk-based maintenance (RBM)[12ndash16] and condition-based maintenance (CBM) [8ndash11]Within the development of a CBM plan failure prognosis isof prominent importance thus there is an ongoing effort oncondition monitoring and calculation of the remaininguseful life [17ndash19] Zeng and Zio [18] developed a dynamicrisk assessment framework based on a Bayesian model to

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5592325 15 pageshttpsdoiorg10115520215592325









1113938θf(x|θ)π0(θ)dθ (1)




π0(θ) 1113946emptyπ1(θ|φ)π2(φ)dφ (2)




(3)



SSE(θ) 1113944n


2 (4)



2 Methodology







f(x|λ) (λt)

xe

minus λt

x x 0 1 (5)



eminus βλ

Γ(α) (6)





(7)

f(t) λeminus λt

(8)


L(λ) 1113945n

i1λe

minus λTi λne

minus λ1113936n

i1 Ti( 1113857 λn

e(minus λnT)

(9)


1113954λ 1T

n

1113936ni1 Ti

(10)


y ax + b (11)



λ (12)


SSE(λ) 1113944n

i1ti minus


λ1113888 1113889

2

(13)


F ti( 1113857 i minus 03n + 04





1 Data collection









5 Result comparison









PilotFilter





Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O


8 9









MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References



































































1113938θf(x|θ)π0(θ)dθ (1)




π0(θ) 1113946emptyπ1(θ|φ)π2(φ)dφ (2)




(3)



SSE(θ) 1113944n


2 (4)



2 Methodology







f(x|λ) (λt)

xe

minus λt

x x 0 1 (5)



eminus βλ

Γ(α) (6)





(7)

f(t) λeminus λt

(8)


L(λ) 1113945n

i1λe

minus λTi λne

minus λ1113936n

i1 Ti( 1113857 λn

e(minus λnT)

(9)


1113954λ 1T

n

1113936ni1 Ti

(10)


y ax + b (11)



λ (12)


SSE(λ) 1113944n

i1ti minus


λ1113888 1113889

2

(13)


F ti( 1113857 i minus 03n + 04





1 Data collection









5 Result comparison









PilotFilter





Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O


8 9









MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References




























































π0(θ) 1113946emptyπ1(θ|φ)π2(φ)dφ (2)




(3)



SSE(θ) 1113944n


2 (4)



2 Methodology







f(x|λ) (λt)

xe

minus λt

x x 0 1 (5)



eminus βλ

Γ(α) (6)





(7)

f(t) λeminus λt

(8)


L(λ) 1113945n

i1λe

minus λTi λne

minus λ1113936n

i1 Ti( 1113857 λn

e(minus λnT)

(9)


1113954λ 1T

n

1113936ni1 Ti

(10)


y ax + b (11)



λ (12)


SSE(λ) 1113944n

i1ti minus


λ1113888 1113889

2

(13)


F ti( 1113857 i minus 03n + 04





1 Data collection









5 Result comparison









PilotFilter





Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O


8 9









MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References




























































(7)

f(t) λeminus λt

(8)


L(λ) 1113945n

i1λe

minus λTi λne

minus λ1113936n

i1 Ti( 1113857 λn

e(minus λnT)

(9)


1113954λ 1T

n

1113936ni1 Ti

(10)


y ax + b (11)



λ (12)


SSE(λ) 1113944n

i1ti minus


λ1113888 1113889

2

(13)


F ti( 1113857 i minus 03n + 04





1 Data collection









5 Result comparison









PilotFilter





Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O


8 9









MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References

































































PilotFilter





Medium-pressuregas

High-pressuregas

F

F

F

4

10

53

2

1

76

M O


8 9









MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References

































































MTTF 1λ (15)





4 Discussion









α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References





























































α β

λ1 λ2 λn

X1 X2 Xn








25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References





























































25

20

15

10

05

00

P (a

lpha

)

00 05 10 15Alpha

(a)

P (b

eta)

4e ndash 05

3e ndash 05

2e ndash 05

1e ndash 05

0e + 050 20000 40000 60000 80000

Beta

(b)


00 05 10 15Alpha

70000

50000

30000

100000

Beta


[1][2]

[3][4]

[5][6]

[7][8]

[9][10]

[11][12]

[13][14]

[15]168E ndash 5








10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References






























































10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 82060h



10e ndash 05

60e ndash 06

20e ndash 06Prob

abili

ty d

ensit

y fu

nctio

n

MTTF = 78219h











18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References

































































18

16

14

12

10

08

Dim

ensio

nles

s MTT

F

PR

Pilo

t

Filte

r

PTG

Calc

ulat

or

Met

er

RCS

THT

tank

THT

pipe

Pum

p

Boile

r

WP

HBDMLELSE







Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References































































Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Pressure regulator

2000 3000 5000400010000Time (days)

00

02

04

06

08

RealHBM

MLELSE

(a)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

Filter

00

01

02

03

04

05

06

07

0 600100 300 400200 500Time (days)

RealHBM

MLELSE

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n Pilot

1000 30001500 250020006000Time (days)

00

02

04

06



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References



























































RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 600 1000400 8000Time (days)

00

01

02

03

04

05

06

07PTG

(a)

0 1500500 1000 2000Time (days)

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

01

02

03

04

05

06

07

RealHBM

MLELSE

RCS

(b)


RealHBM

MLELSE

70002000 50003000 400010000Time (days)

00

02

04

06

08

6000

THT tank

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

0 2000 4000 6000 8000 10000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

RealHBM

MLELSE

THT pipe

(b)


RealHBM

MLELSE

Calculator

3000 50002000 400010000Time (days)

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)

Meter

300025001000500 15000 2000Time (days)

RealHBM

MLELSE

00

02

04

06

08

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(b)



RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References



























































RMSE


2

n

1113971

(16)


5 Conclusions



RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

12000 140004000 100002000 6000 80000Time (days)

00

02

04

06

08

Water pipe

(a)

0 500 1000 1500 2000Time (days)

RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

00

02

04

06

Boiler

(b)


RealHBM

MLELSE

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

200 1000400 600 1200800 14000Time (days)

00

02

04

06

08




Data Availability




References




























































Data Availability




References



























































Leoni, Leonardo; Cantini, Alessandra; Bahootoroody ...

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Transcript of Leoni, Leonardo; Cantini, Alessandra; Bahootoroody ...