Research Article A Study on Estimating the Next Failure

Research ArticleA Study on Estimating the Next Failure Time ofCompressor Equipment in an Offshore Plant

SangJe Cho1 Jong-Ho Shin2 Hong-Bae Jun3 Ho-Jin Hwang4

Chunghun Ha3 and Jinsang Hwang5

1Department of Mechanical Engineering EPFL (SCI-STI-DK) Lausanne Switzerland2Department of Industrial Engineering Chosun University Kwangju Republic of Korea3Department of Industrial Engineering Hongik University Seoul Republic of Korea4Korea Research Institute of Ships and Ocean Engineering Daejeon Republic of Korea5PartDB Co Daejeon Republic of Korea

Correspondence should be addressed to Hong-Bae Jun hongbaegmailcom

Received 23 June 2016 Accepted 3 October 2016

Academic Editor Inmaculada T Castro

Copyright copy 2016 SangJe Cho 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

The offshore plant equipment usually has a long life cycle During its OampM (Operation and Maintenance) phase since theaccidental occurrence of offshore plant equipment causes catastrophic damage it is necessary to make more efforts for managingcritical offshore equipment Nowadays due to the emerging ICTs (Information Communication Technologies) it is possible tosend health monitoring information to administrator of an offshore plant which leads to much concern on CBM (Condition-Based Maintenance) This study introduces three approaches for predicting the next failure time of offshore plant equipment (gascompressor) with case studies which are based on finite state continuous time Markov model linear regression method and theirhybrid model

1 Introduction

In general maintenance is defined as all technical and man-agerial actions taken during usage period to maintain orrestore the required functionality of an asset or equipmentThere have been various classifications of maintenance poli-cies Simply maintenance policies can be divided into break-down maintenance (corrective maintenance) preventivemaintenance and CBM (Condition-Based Maintenance)Unlike breakdownmaintenance and preventivemaintenancethe CBM focuses on not only fault detection and diagnosticsof equipment but also degradation monitoring and failureprediction Generally CBM can be treated as a method usedto reduce the uncertainty of maintenance activities and iscarried out according to the requirements indicated by equip-ment condition [1] Thus the CBM enables us to identifyand solve problems in advance before equipment damageoccurs In industry systems any equipment damage canlead to serious results Since a critical failure or degrada-tion of the equipment during its operation can seriously

damage the belief of customers on the equipment reliabilitythe maintenance enhancement for preventing this kind offailure or degradation in advance has precedence over anyother things in a company Since oil and gas industriesare particularly capital-intensive careful management onequipment is very important In this respect the CBM is avery attractive method for oil and gas industries CBM iscurrently being utilized in the petrochemical industry withcondition monitoring of both on and offshore oil and gaswells [2]

Until now it has been difficult to achieve the effectivenessof maintenance operations because there is no informa-tion visibility during equipmentrsquos usage period Howeverrecently with emerging technologies such as various sen-sors SCADA (Supervisory Control And Data Acquisition)and PEID (Product Embedded Information Devices) areexpected to be rapidly used for gathering and monitoringthe status data of equipment during equipment usage periodAdvancements in information communication technologyhave added accelerated growth in the CBM technology area

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8705796 14 pageshttpdxdoiorg10115520168705796

2 Mathematical Problems in Engineering

by enabling network bandwidth data collection and retrievaldata analysis and decision support capabilities for large datasets of time series data [3] Under the new environmentwe can gather the equipment status data related to usageconditions failure maintenance or service events and so onThese data sets enable us to diagnose the degradation statusof the equipment in a more exact way Therefore using thisinformation gives us new challenging issues for improvingthe efficiency of equipment maintenance operations We candiagnose equipment status predict equipment abnormalityand execute proactive maintenance that is doing CBM

Although there have been some relevant research worksso far the CBM is still challenging area Current approacheshave the limitation in detailed methods or validated predic-tive models in particular in the offshore plant domain Thisstudy deals with the approaches that can predict the next fail-ure time of offshore plant equipment (gas compressor) usedin LNG FPSO (Liquefied Natural Gas Floating ProductionStorage and Offloading vessel)

An LNG FPSO is an offshore plant of delivering liquefiedgas from a gas field to customers Recently the demand forLNG FPSO is highly increasing and the demand for LNGFPSO projects will grow along with the increased demandfor natural gas [4] The OampM phase of LNG FPSO requiresheavy charges and more efforts to optimize the cost and toreduce the risks than the construction phase because of itslong life cycle Nowadays due to the fact that an accident ofLNG FPSO in operation causes catastrophic damage manystudies have focused on a maintenance system In this veinthis study focuses on the prognostic approach for the gascompressor equipment in LNG FPSO which is one of themain results for the Korean government supported projectthat is being currently developed since 2013 with the objectiveof implementing the predictive maintenance system for LNGFPSOsThe objective of this study is to develop the algorithmfor estimating the next failure time of a gas compressorbased on gathered vibration sensing and failure data Tothis end in this study finite state Markov model based ap-proach linear regression model based approach and theirhybridmodel have been introduced To evaluate the proposedapproaches the case study and computational experimentsfor compressor equipment have been carried out

The rest of this study is organized as follows First rele-vant previous studies are reviewed and their limitations arediscussed in Section 2 Then Section 3 describes the equip-ment for CBM focused on this study and two approachesfor estimating the next failure time of a compressor andSection 4 introduces relevant case studies and computationalexperiments In Section 5 the hybrid approach combiningtwo approaches is proposed with computational experimentFinally this study is concluded with further research issues inSection 6

2 Literature Review

There are several maintenance policies corrective mainte-nance preventive maintenance opportunistic maintenanceCondition-Based Maintenance and predictive maintenance

Correctivemaintenance is the unplannedmaintenanceHow-ever preventivemaintenance (such as constant intervalmain-tenance age basedmaintenance and imperfectmaintenance)and predictive maintenance (such as RCM (Reliability Cen-tered Maintenance) and CBM) are the types of plannedmaintenance Formore details please refer to Bevilacqua andBraglia [5] or Kothamasu et al [6] Among various mainte-nance policies this study focuses on the CBM

The term CBM is often used with other terms suchas PdM (Predictive Maintenance) PHM (Prognostic andHealth Management) and on-condition maintenance whichcomes from the US Department of Defense and Departmentof Energy In this study we define CBM as a maintenancepolicy which does maintenance action before equipmentfailures happen by assessing equipment condition includingoperating environments and predicting the risk of equip-ment failures in a real-time way based on gathered dataThe benefits of a successful CBM strategy are expected toinclude less regular maintenance the reduction of unsched-uled maintenance and improved supply chain management[7]

Until so far there have been several research works aboutCBM For example Lee [8] introduced the fundamentaltechnologies for remotemaintenance (called teleservice engi-neering) and CBM machine performance assessment andremote diagnosis Lee [9] introduced a new methodology ofCBM called machinery dynamics and data fusion throughremote machinery monitoring He presented an example ofa remote wireless application currently in use for monitoringmachinery in industrial plants Dieulle et al [10] dealt withthe problem related to CBM policy for a single-unit deteri-orating system and proposed the approach to determine theoptimal inspection schedule and replacement threshold withrenewal processes theory Furthermore Grall et al [11] dealtwith a condition-based inspectionreplacement problem fora stochastically and continuously deteriorating single-unitsystem With regenerative and semiregenerative processestheory they tried to find twomaintenance decision variablespreventive replacement threshold and inspection schedulewith the objective of minimizing the long run expectedmaintenance cost per unit time In addition Lin and Tseng[12] combined traditional reliability modelling methods withvibration-based monitoring techniques and artificial neuralnetwork technologies in an integrated system to determinethe health status of machinery namely CMAC-PEM (Cere-bellar Model Articulation Controller neural network-basedmachine Performance Estimation Model) They developeda WPHM (Weibull Proportional Hazards Model) and car-ried out a bearing deterioration experiment to test boththe CMAC-PEM and the WPHM Moore and Starr [13]have reviewed the methods and functionality of criticalityassessments in condition-based monitoring and proposedthe CBC (Cost Based Criticality) algorithm to rank all thealarms arising from conditionmonitoring which could allowoptimized prioritization of maintenance activities Wu etal [14] proposed an intelligent decision support system forthe optimal CBM policy They developed a neural networkmodel that uses bearing vibration information to predict thelife percentage of a machine and the remaining life of the

Mathematical Problems in Engineering 3

machineThey also computed the optimal replacement strate-gies by proposing a cost matrix method and suggested anoptimal replacement time for assistingmaintenance decision-making In addition in their other work [15] they proposeda prognostic method for machine degradation tracking usingARIMA (AutoRegressive Integrated Moving Average) timeseries model in order to predict the pending failures and theRUL (Remaining Useful Life) of machines They developed aforecasting strategy and an automatic prediction algorithm ofARIMAmodels and carried out the analysis on the vibrationseverity data collected from rotating machines They com-pared the performance of the proposed approach with thebasic Box-JenkinsARIMAmethod RecentlyHashemian andBean [16] discussed the limitations of time-based equipmentmaintenance methods and the advantages of predictive oronline maintenance techniques in identifying the onset ofequipment failure Gruber et al [17] suggested a CBM frame-work that is based on system simulations and a targetedBayesian network model Simulations are used for explor-ing various CBM policies under different scenarios and theBayesian network is used for failure prediction based onsimulation dataTheproposed framework has been applied toa freight rail fleet case In addition Lee et al [18] carried outa comprehensive review of the PHM field They introduceda systematic PHM design methodology 5S methodologyfor converting data to prognostics information They alsopresented a systematic methodology for conducting PHM asapplied to machinery maintenance

In particular some research works focused on Markov-based model for CBM For example Bunks et al [19] intro-duced an application of HMMs (Hidden Markov Models) toCBM for the helicopter gearbox case and presented examplesof torque level defect level and defect-type classificationThey concluded that HMMs have a strong potential forconstructing practical and robust algorithms for CBM Fur-thermore Ambani et al [20] developed a continuous timeMarkov chain degradation model and a cost model to quan-tify the effects of maintenance on a multiple machine systemTo evaluate the effectiveness of the proposed methods theyintroduced a case study of an automotive assembly lineIn their model a Markov-based degradation model is usedto represent discrete state degradation process under theassumption that the future degradation state of the machinedepends only on the current degradation state and not on thehistory of the degradation states Si et al [21] mentioned thatMarkovian-based models have been widely applied to RULestimation and to maintenance decision-making supportThey said that themain reason forMarkovian-basedmodel isthat the plant operation condition can be divided into severalmeaningful states such as Good OK andMinor defects onlyMaintenance required and Unserviceable so that the statedefinition is closer to what is used in industry than otherstochastic models and therefore is easy to understand

On the other hand some previous works have focusedon the maintenance of offshore plant For example Wangand Majid [22] carried out a case study of the analysis ofreliability and maintenance data on five gas turbines andof the modelling for determining the appropriate preventivemaintenanceinspection intervals of offshore oil platform

plant Arthur and Dunn [23] introduced an application of anoptimized CBM approach to large reciprocating compressorson an offshore installation Caselitz and Giebhardt [24] intro-duced results of work in the field of condition monitoringand fault prediction in offshorewind energy convertersTheirwork included not only development hardware and softwaresolution but also prototype tests and integration of fault pre-diction and maintenance and repair scheduling techniquesFurthermore Dey et al [25] developed a risk based main-tenance model using a combined multiple-criteria decision-making and weight method for offshore oil and gas pipelinesEleye-Datubo et al [26] and Eleye-Datubo et al [27] appliedBayesian network methods for examining the system safetyof FPSOs Miguelanez and Lane [28] presented the recoverysystem of offshore wind turbines The recovery system takesa broad view of events and sensor values across the completeturbine system and subsystems In addition Hussin et al[29] dealt with a systematic methodology for analyzing themaintenance data of gas compression train system on anoffshore platform to gain insight about the system reliabilityperformance and identify the critical factors influencing theperformance Telford et al [2] explored the existing literatureon the development and applications of CBM in the oiland gas industry de Andrade Melani et al [30] proposed amethod for risk analysis of LNG carriers operations basedon Bayesian network method Recently Griffith et al [7]addressed initial development and integration of SHPM(StructuralHealth andPrognosticsManagement) system intothe OampM process for offshore wind power plants Theydeveloped a multiscale simulation-based methodology toinvestigate the sensitivity (or effects) of damage of bladesCho et al [31] proposed a linear regression-based approachfor estimating the remaining life time of compressor Choet al [32] reviewed previous studies associated with CBMof offshore plants and introduced case studies of prognosissystem development predicting performance and failures ofoffshore plant equipment such as compressor and pumptower

Although not a few previous research works dealt withvarious CBM issues little attention has been paid to theresearch that deals with offshore plant equipment and has thelimitation in estimating the next failure time (remaining lifetime from the current time) based on gathered sensor dataEstimating the next failure time with sensor data is still theundeveloped area in an offshore plant equipment To copewith the limitations this study proposes three approaches toestimate the next failure time of offshore equipment based onfinite state continuous time Markov model linear regressionmodel and their hybrid model

3 Target Equipment andProposed Approaches

LNG FPSOs are used when an oil platform is in a remote ordeepwater location where seabed pipelines are not cost effec-tive [33] Nowadays due to the fact that an accident of LNGFPSO in operation causes catastrophic damage many studiesdealt with the improvement of operating a maintenance


Produced waterconditioner

HP separator

Compressor

Surge tank

Pump

Gas dehydration

Good fluid

Producedwater

Oil

To flare

Gas

Produced water

Overboard

Oil

Export gas

Lift gas

Oil exportMain oil line

Figure 1 Process flow diagram of water oil and gas separation in LNG FPSO

system for LNG FPSO The LNG FPSO is composed of lotsof facilities and equipment Depending on location they canbe classified into top-side hull-side and subsea-side Amongthem this case study focuses on a gas compressor in the top-side of LNGFPSO Figure 1 shows the process flowdiagramofwater oil and gas separation in LNG FPSO The compressoris used in liquefaction processes of offshore plants It isimportant equipment in not only offshore but also onshoreplants It is a mechanical device to increase the pressure ofgas and to reduce its volume It spends most of the energyin offshore plants Offshore gas compressors are used forvarious tasks including reservoir management productionenhancement and the transmission of gas Any unexpectedor prolonged downtime of these units has a large impacton plant availability as a loss of compression capabilitydrastically affects the oil and gas production of the asset[23] There are several kinds of compressors Among themthis study deals with the centrifugal compressor Accordingto Hussin et al [29] the number of failures of centrifugalcompressor used in an offshore plant is about 26 duringsix years For this reason it preferentially needs to do thedevelopment of a prognosis system for the compressor

According to OREDA [34] frequently observable failuremodes of compressor are low output overheating spuriousstop external leakage and so on The failure mode is usuallygenerated from some failure causes Compressor failurecauses (frequency) are shown as follows

Rotationshaft (22) instrumentation (21) radialbearing (13)Bladeimpeller (8) thrust bearing (6) compressorseal (6)Motor winding (3) diaphragm (1) and so forth(20)

Hence in this study we focus on one main causerotationshaft for CBM

31 Vibration Analysis Compressor There are some kindsof parameters to monitor the status of the compressor Thisstudy deals with a vibration parameter because it is widelyused in detecting the status of rotating equipment Generallyspeaking vibration is the value with time of the magnitudeof a quantity that is descriptive of the motion or position of amechanical system Because most normal plant equipment ismechanical vibration monitoring provides the best tool forroutine monitoring and identification of incipient problems[35] The increasing amplitude of vibration may be anindication of a deteriorating machine condition and the rateof increase is proportional to the degree of damageThereforeit is possible to predict the trend of deterioration of amachineby monitoring the amplitudes of its fault related vibrationfeatures [36]

Relative shaft vibration and bearing vibration data areusually used to evaluate the status of a compressor of an LNGFPSO In this study wemonitor the status of a gas compressorthrough relative shaft vibration data ISO 7919 (internationalstandard for relative shaft vibration of rotating machines)suggests a way for the measurement of a vibration parameterAccording to ISO 7919 the max value among two peak-peakvalues measured by two sensors located in 119909-axis and 119910-axisas 90 degree is expressed as 119878(p-p)max = [119878119883(p-p) 119878119884(p-p)]max

A common practice in industry is to set up various warn-ing levels instead of maintenance stages The warning levelscan be classified as alert high alert alarm serious alarmand breakdown General guidelines for setting up warninglevels for different types of machines are recommended byvarious national and international committees [36] ISO7919 recommends four vibration limits limit of start-upperformance (A) limit of good vibration performance (B)limit for warning alarm (C) and limit for trip (D) Thevalues of limits are calculated as followsAB (4800radic120587) BC(9000radic120587) and CD (13200radic120587) where 120587 denotes the RPM(Revolution Per Minute) of a compressor


Relative shaft vibration data on a gas compressor

Markov model based approach

(1) Collecting relative shaft vibration data classified by vibration state

levels

(2) Modeling continuous time Markov chain and estimating

state transition rates

(3) Estimating the next failure time based on state transition

diagram analysis

The next failure time

Regression model based approach

(1) Calculating the movingaverage of vibration data

(2) Finding the abnormal point by comparing vibration data with

filtered data

(3) Estimating the next failuretime by simple linear

regression model


Hybrid approach

YesNo

Vibration level gtthreshold value

Figure 2 Overview of proposed approaches

In this study we propose three approaches for estimatingthe next failure time of a compressor (1) Markov modelbased approach (2) regression model based approach and(3) hybrid approach combining (1) and (2) In the Markovmodel based approach we divide the vibration value intosome levels and stochastically predict the next failure timebased on the finite state continuous time Markov modeltheory In the regression model based approach when thevibration data evidently differs from themoving average filterand shows increasing trends we predict the next failure timeusing simple linear regression model The hybrid approachtakes the advantage of two approaches Depending on thevibration level one of two approaches is applied to estimatethe next failure time Figure 2 depicts the detailed processflow of these approaches

32 Markov Model Based Approach In order to apply theCBM policy of an LNG FPSO compressor among variousprognostics methods such as wavelets artificial neural net-works and Bayesian network in this study we use the finitestate continuous time Markov model since it has the benefitin the amount of data required for analysis compared toother methods Furthermore it allows an exact computationof system reliability

In this study it is assumed that LNG FPSO operationsystem records 119878(p-p)max and its timestamp data wheneverthe level of relative shaft vibration exceeds the predefinedlimit (eg warning level and trip level) Furthermore theamplitude of vibration signal of the compressor will remainthe limit of tolerance range unless it has abnormal symptomsfor faults or failures In addition we assume that the vibrationstate evolves continuously over time and state transition ratedoes not depend on time based on the interview result withcompressor engineers

To maintain brevity and consistency this study definesthe following notation

119904 index of compressor state1205911199041199041015840 the number of changes from 119904 to 11990410158401198791199041199041015840 transition rate from 119904 to 11990410158401199051198991199041199041015840 the time interval of the 119899th status transition fromstatus 119904 to status 1199041015840119871 1199041199041015840 the expected time to reach the state 1199041015840 from thestate 1199041198641199041199041015840 the expected time from status 119904 to the very nextstate 1199041015840119875119903119904 the probability to transit from state 119904 to state 119904+1119875119895119904 the probability of the 119895th return to the state 119904 aftergoing through state 119904 minus 1120587 the RPM (Revolution Per Minute) of a compressor120582 the deteriorating transition rate120583 the recovery transition rate

The detailed procedures of the proposed approach are asfollows

Step 1 (collecting relative shaft vibration data classified byvibration state levels) We assumed that LNG FPSO opera-tion system records 119878(p-p)max and its timestampdatawheneverthe level of relative shaft vibration exceeds the predefinedlimit Regarding the predefined limit based on ISO 7919 weset the vibration levels into good vibration level and alarmlevel Andwe divide the alarm level into lowmiddle and highlevels in detail because it is more important to predict thenext failure time in an alarm level than in a good vibration


Event occurrence

Operation start(Unit time)

State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax

Figure 3 Example of vibration state transition plot in Markovmodel based approach

performance levelThus whenever the vibration data exceedsover the predefined levels the state values (denoted as 119904) forrepresenting each level and its timestamp data are recordedThe value of state 119904 has the following meaning

119904 = 0 a good vibration level under 9000radic120587119904 = 1 a low alarm level under 10400radic120587119904 = 2 a middle alarm level under 11800radic120587119904 = 3 a high alarm level under 13200radic120587119904 = 4 a trip level above 13200radic120587

Figure 3 depicts an example of transition plot represent-ing the state values of vibration data changed over time

Step 2 (estimating status transition rates) From the interviewwith compressor maintenance engineers we identified thatthe failures of the compressor shaft are not much related tothe deterioration process and occur randomly Hence in thisstudy we assume that the time when status transition occursfollows the homogeneous Poisson process having the statetransition rate as follows

1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)

Then since the state transition rate is constant and doesnot depend on the time we assume that it has the Markovianproperty

Step 3 (estimating the next failure time) Then 1198710119904 could beestimated by the following formula

1198710119904= infinsum119895=0

[119875119895119904minus1 sdot 119875119903119904minus1 sdot 119895 sdot (119871 119904minus2119904minus1 + 119864119904minus1119904minus2) + 119864119904minus1119904]+ 1198710119904minus1

(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4

Let 119895 be the number of first passage from 1199041015840 minus 1 to 1199041015840 minus 2The first term of formula (2) indicates the expected time untilarriving at state 119904 after the 119895th return to state 119904minus1 calculated bymultiplying the duration time that is (119895sdot(119871 119904minus2119904minus1+119864119904minus1119904minus2)+119864119904minus1119904) and its probability (119875119895119904minus1 sdot 119875119903119904minus1) The second term (ie1198710119904minus1) of formula (2) denotes the expected time from state 0to state 119904 minus 1

Let 120582119904 = 119879119904119904+1 0 le 120582119904 le 1 for 119904 = 0 1 2 3 whichindicates the deteriorating transition rate and 120583119904 = 119879119904+11199040 le 120583119904 le 1 for 119904 = 0 1 2 3 which indicates the recoverytransition rate respectively Then according to Anderson[37] 119864119904minus1119904 = 1120582119904minus1 119864119904119904minus1 = 1120583119904 119875119895119904 = (120583119904(120582119904 + 120583119904))119895 and119875119903119904minus1 = (120582119904(120582119904 +120583119904)) in the continuous timeMarkov modelAs a result formula (2) could be unfolded as follows

1198710119904 = infinsum119895=0

[119875119895119904minus1 sdot 119875119903119904minus1sdot 119895 sdot (119871 119904minus2119904minus1 + 119864119904minus1119904minus2) + 119864119904minus1119904] + 1198710119904minus1= infinsum119895=0

[( 120583119904minus1120582119904minus1 + 120583119904minus1)119895 sdot ( 120582119904minus1120582119904minus1 + 120583119904minus1)

sdot 119895 sdot (119871 119904minus2119904minus1 + 1120583119904minus1) +1120582119904minus1] + 1198710119904minus1

= infinsum119895=0


sdot 119895 sdot (119871 119904minus2119904minus1 + 1120583119904minus1)] +infinsum119895=0

[( 120583119904minus1120582119904minus1 + 120583119904minus1)119895

sdot ( 120582119904minus1120582119904minus1 + 120583119904minus1) sdot1120582119904minus1 ] + 1198710119904minus1

= infinsum119895=0


sdot (119895 + 1) sdot (119871 119904minus2119904minus1 + 1120583119904minus1)]

minus infinsum119895=0


sdot (119871 119904minus2119904minus1 + 1120583119904minus1)] +infinsum119895=0


sdot ( 120582119904minus1120582119904minus1 + 120583119904minus1) sdot1120582119904minus1 ] + 1198710119904minus1 = (

120582119904minus1120582119904minus1 + 120583119904minus1)

sdot (119871 119904minus2119904minus1 + 1120583119904minus1) sdotinfinsum119895=0

[( 120583119904minus1120582119904minus1 + 120583119904minus1)119895 sdot (119895 + 1)]

minus (119871 119904minus2119904minus1 + 1120583119904minus1) +1120582119904minus1 + 1198710119904minus1

(3)


Time

13200 radic120587 Regression line

Moving averagefilter

The next failure time120578 value

S (p-

p)m

ax

S(p-p)max

Figure 4 Example of the trend plot of vibration data in regression model based approach

Sinceinfinsum119895=0

[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)

where 119870 = 120583119904minus1(120582119904minus1 + 120583119904minus1) the above equation could beexpressed as follows

= ( 120582119904minus1120582119904minus1 + 120583119904minus1) sdot (119871 119904minus2119904minus1 +1120583119904minus1)

sdot 1[1 minus (120583119904minus1 (120582119904minus1 + 120583119904minus1))]2 minus (119871 119904minus2119904minus1 +

1120583119904minus1)+ 1120582119904minus1 + 1198710119904minus1

= (120582119904minus1 (120582119904minus1 + 120583119904minus1)) minus [1 minus 120583119904minus1 (120582119904minus1 + 120583119904minus1)]2[1 minus 120583119904minus1 (120582119904minus1 + 120583119904minus1)]2

sdot (119871 119904minus2119904minus1 + 1120583119904minus1) +1120582119904minus1 + 1198710119904minus1

= (120582119904minus1 sdot 120583119904minus1) (120582119904minus1 + 120583119904minus1)2[1 minus 120583119904minus1 (120582119904minus1 + 120583119904minus1)]2 sdot (119871 119904minus2119904minus1 + 1120583119904minus1)+ 1120582119904minus1 + 1198710119904minus1

= 120583119904minus1120582119904minus1 sdot (119871 119904minus2119904minus1 +1120582119904minus1) +

1120582119904minus1 + 1198710119904minus1

(5)

Along formula (5) it is possible to estimate11987104 as follows11987104 = [12058331205823 sdot 11987123 +

11205833] +11205823 + 11987103 (6)

Also we can know that the expected time to reach state 119904from state 119904 minus 1 is simply calculated as follows

119871 119904minus1119904 = 1198710119904 minus 1198710119904minus1 (7)

Along formulae (5) and (7) 11987103 and 11987123 in formula (6)could be estimated as follows

11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)

1198710119904 and 119871 119904minus1119904 in a series of calculations like 11987102 and 11987112in formula (8) could be estimated using formulae (5) and (7)in the same way

If the current state is 119904 then 119871 1199044 could be estimated asfollows

119871 1199044 = 11987104 minus 1198710119904 (10)

Then finally the next failure time could be estimated asfollows

The next failure time = the current time + 119871 1199044 (11)

33 Regression Model Based Approach In addition to theMarkov model based approach in this study the regressionmodel based approach for estimating the next failure timeof the compressor is proposed The regression model basedapproach is based on the 119878(p-p)max trend plot (please refer toFigure 4)The trend plot is amethod to record the variation ofthemagnitude that is descriptive ofmotion or positions of theequipment with time The trend plot helps engineers figureout the status of the equipment at a glance After the raw dataof vibration is recorded in the trend plot the regressionmodelbased approach analyzes 119878(p-p)max trend plot with movingaverage filter and abnormal indicator variable (denoted as V119894)According to ISO 7919 we let the limit for trip be a criticallimit for the failure of a compressor Then it calculates thepoint where the simple linear regression line intersects withthe limit for trip and considers it as the next failure time


119896 the number of 119910119894rsquos119898 index for calculating 120578


119899 the number of entities used in the moving averagefilter119903 the number of comparisons with 119910119894 and 119910119894V119894 the 0-1 binary variable that indicates whether 119910119894 lt119910119894 or not (ie whether abnormal situation occurs ornot)119909119894 the time when the 119894th 119878(p-p)max value (relative shaftvibration data) is recorded119910119894 the 119894th 119878(p-p)max value119910119894 moving average filter of 119910119894120578 the time point just before the vibration value iscontinuously over themoving average filter valuewith119903 times120587 the RPM (Revolution Per Minute) of a gas com-pressor

The detailed procedure is as follows

Step 1 (calculating the moving average of vibration data)In this study in order to predict the next failure time it isnecessary to carefullymonitor the trend of vibration data overtime points Since the vibration value itself has the limitationin giving the trend of time series values the moving averagefilter is applied to catch the trend of vibration values If thereare 1199101 1199102 119910119896 a moving average filter 119910119894 for 119878(p-p)max iscalculated as follows

119910119894 = 119910119894minus119899+1 + 119910119894minus119899+2 + sdot sdot sdot + 119910119894119899 119899 le 119894 le 119896 (12)

Step 2 (finding the abnormal time point by comparing vibra-tion datawith filtered data) If there is the time point showingabnormal situations continuously compared to previous datawe assume that the failure propagation evolves in a fast wayafter that point In this study to find the abnormal time point120578 is calculated by formula (13)Here 120578 indicates the timepointjust before showing the abnormal performance (ie 119910119894 lt 119910119894)continuously in 119903 times

120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where

V119894 = 1 (if 119910119894 lt 119910119894)0 (otherwise) for 119899 le 119894 le 119896 (14)

Step 3 (estimating the next failure time by a linear regressionmodel) The next failure time could be estimated based onthe values of 119909120578+1 119909120578+2 119909119896 by a linear regression modelIf the regression equation is expressed as119910119894 = 0+1 sdot119909119894 where0 = 119910minus1 sdot119909 and 1 = sum119896119894=120578+1(119909119894minus119909)(119910119894minus119910)sum119896119894=120578+1(119909119894minus119909)2then the next failure time could be estimated by formula (15)

The next failure time = 11205731 sdot (

13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax

Figure 5 Example of vibration data

4 Case Study

For a case study we used the data on shaft vibration ofa gas compressor generated based on the sample vibrationdata obtained from a compressor manufacturing companyin South Korea The generated data for about three yearsincludes 20 failures and about 564 state transitions whereRPM is 3600 (ie 120587 = 3600) The vibration data used in thisstudy is depicted by Figure 5

41 Case Study of Markov Model Based Approach From theexample data of the case study we obtain the number ofchanges from 119904 to 1199041015840 that is 1205911199041199041015840 as follows 12059101 = 13512059110 = 115 12059112 = 108 12059121 = 88 12059123 = 59 12059132 = 39 and12059134 = 20 Here we describe how to calculate the transitionrate 1205823 that is 11987934 Other calculations for 1198791199041199041015840 are omittedfor the convenience1205823 could be estimated as follows

1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)

We could obtain the transition rates in the same way asfollows 1205820 = 00068 1205821 = 00506 1205822 = 00569 1205831 = 005801205832 = 00453 and 1205833 = 00392 The state transition diagramwith these transition rates is depicted in Figure 6

Where the current state is the state 0 the expected timeto reach the failure from the state 0 (11987104) could be estimatedalong formula (6) as follows

11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)

In formula (17)11987123 and11987103 could be estimated in a seriesof calculations as follows

11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102


State(0) Good vibration performance(1) A warning alarm (low)(2) A warning alarm (middle)(3) A warning alarm (high)(4) A trip

1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4

Figure 6 Example of state transition diagram

11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)

Table 1 shows a result of a series of the above formulaeThe time point when we estimated the next failure time is27628 (hours) Since 119878(p-p)max value is 1978346 at that time thevibration state is the state level 3Thus the next failure time isas follows the current time point + 11987134 = 27628 + 13417 =2776217 (hours)The real failure time is known as 27776Theresidual life time at 27628 time point is only about 134 (hours)Hence we could see that the estimated next failure time byMarkov model based approach is close to the real one

42 Computational Experiments of Markov Model Based Ap-proach In order to evaluate the performance of the Markovmodel based approach we have carried out computationalexperiments based on the vibration history data (refer toFigure 5) To quantify the degree of the performance of theproposed approach this study uses the MAPE (ModifiedAbsolute Percentage Error [38]) measure represented asfollows

MAPE = 10038161003816100381610038161003816100381610038161003816119877 minus 119877lowast(119877 + 119877lowast) 2

10038161003816100381610038161003816100381610038161003816 sdot 100 (19)

Table 1 119871 1199041199041015840 calculations119899 1199051198993411987104 693307211987103 559140311987102 357345311987101 147977811987134 134166911987123 201795011987112 2093675

Table 2 Test results of Markov model based approach

Failure event AverageMAPElowast

♯ obs lessthan 30dagger

♯ obs lessthan 50clubs

1 0308 6 82 0310 5 83 0498 2 64 0407 5 65 0653 3 46 0422 5 77 0377 3 58 0667 5 59 0438 6 810 0644 2 511 0485 4 512 0387 5 513 0258 6 10Average 0450 4380 6310lowastThe average on 10 MAPEs for each failure eventdaggerNumber of observations such that MAPE lt 30 among 10 test examplesclubsNumber of observations such that MAPE lt 50 among 10 test examples

where 119877lowast indicates residual time to the next failure and 119877 isthe estimated residual time to the next failure

With MAPE we could avoid the problem of large errorswhen the residual life time to the next failure is close tozero To calculate the value of MAPE in the computationalexperiment we use the data after 10000 hours becauseMarkov model based approach needs enough history dataAnd then we randomly chose ten time points for eachfailure time and estimated the failure time at those timesTable 2 shows that the average MAPE of Markov modelbased approach is 4503Observation results for the numberof solutions less than 30 or 50 showed us that a fewsolutions have highMAPEvalues so that they seemed to affectthe overall performance of the approach Furthermore wecould find thatMarkovmodel based approach has more largeMAPEs as the time point becomes close to the real failuretime

43 Case Study of RegressionModel Based Approach Thecasestudy of regression model based approach has been carried


y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax

Figure 7 Example of regression model based approach

out based on 100 time point data sets as shown in Table 3Note that 119910119894 is the maximum 119878(p-p)max value at time 119894 whoseunit time is an hour We set parameters as follows 119899 = 10119903 = 5 and 119896 = 27705 Table 3 shows moving average filter119910119894 indicator variable V119894 and prod119898+119903minus1119894=119898 V119894 for the time point 119909119894Along formula (13) 120578 is 27676 in this case study

Figure 7 depicts how to estimate the next failure timeThelinear regression model in this case study is calculated as 119910 =07305119909 minus 20066 based on the values where (11990927677 11991027677)(11990927678 11991027678) (11990927705 11991027705) Along formula (15) thenext failure time could be estimated as follows

The next failure time = 107305 ( 13200radic3600 + 20066)≐ 2775633 (hours)

(20)

The real failure time is 27776 Since the time point whenwe estimated the next failure time is 27705 the remaininglife time is 5133 (hours) Hence we could also see thatthe estimated next failure time by regression model basedapproach is close to the real one

44 Computational Experiments of Regression Model BasedApproach In order to evaluate the performance of regressionmodel based approach we have also carried out computa-tional experiments Table 4 shows the values of MAPEs whencomparing the residual life time estimated from proposedregression model based approach to the real residual lifetime According to this table we could find the trend thatregression model based approach has good performance asthe residual life time becomes close to zero compared toMarkov model based approach However we could also seethat the regression model based approach has the limitationin estimating the next failure time when the residual life timeis relatively long since the linear regression model has thetendency in being highly affected by abnormal data

5 Hybrid Approach

Throughout case study and computational experiments fortwo approaches we could find that Markov model basedapproachhasmore better performance than regressionmodelbased approach when the remaining residual life time isrelatively long while regression model based approach hasmore better performance in the opposite case The Markovmodel based approach could have stable performance com-pared to the regression model based approach although itseems to give not good performance as the residual life timebecomes closed to the failure time On the contrary thelinear regression method is straightforward and practicaland it gives us the relative good performance when theresidual life time is closed to the failure time comparedto the Markov model based approach However it is sosensitive on some abnormal data that it often overestimatesor underestimates them For this reason in this study wesuggest the hybrid approach that applies both Markov modelbased approach and regression model based approach intoestimating the next failure time In the hybrid approach untila certain vibration level Markov model based approach isapplied and then after over the certain vibration level theregression model based approach is used to estimate the nextfailure time The below formula shows us how to apply bothapproaches depending on the level of vibration in the hybridapproach

The next failure time = EFT1 (119910119894 lt 120596)EFT2 Otherwise (21)

where EFT1 and EFT2 represent the estimated next failuretime by the Markov model based approach and that bythe regression model based approach respectively and 120596indicates the vibration index

In order to minimize the MAPE it is necessary to findthe best 120596 To this end we have carried out the experimentswith increasing 120596 from 3000radicRPM to 12600radicRPM by


Table 3 Data and variables used in the regression model basedapproach

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1


Table 4 Computational experiment results of regression model based approach

RLTlowast 10 15 20 25Nodagger EstDagger MAPE Est MAPE Est MAPE Est MAPE1 9323 701 18692 2191 29264 3761 45582 58322 9836 165 21562 3589 36439 5825 87836 111383 11632 1509 24609 4852 38227 6261 58742 80594 6703 3947 13807 828 22950 1374 37241 39335 4897 6852 10003 3997 17013 1614 26157 4526 10510 497 21993 3781 35793 5661 59961 82307 9318 706 18093 1869 27484 3152 40181 46588 6443 4326 14471 359 26705 2871 181786 151649 10946 903 22764 4112 36031 5722 56923 779310 9669 336 18323 1994 27370 3112 39112 440211 9845 157 20124 2917 31880 4580 50038 667312 6977 3562 14004 687 22399 1132 33762 298213 7267 3165 14912 059 23800 1735 36211 3663Average 2064 2403 3600 6383Total average 3612lowastResidual life time (unit hour)daggerFailure event numberDaggerResidual life time estimated by regression model based approach (unit hour)

Table 5 Test results depending on 120596120596 Average MAPE3000radicRPM 48643600radicRPM 47884200radicRPM 47044800radicRPM 45855400radicRPM 43936000radicRPM 42136600radicRPM 40057200radicRPM 38027800radicRPM 37258400radicRPM 37099000radicRPM 37399600radicRPM 373210200radicRPM 378510800radicRPM 379811400radicRPM 382712000radicRPM 384812600radicRPM 3869

600radicRPM Table 5 shows the result of experiments depend-ing on the changes of 120596 It shows us that 120596 = 8400radicRPMhas the best performance

6 Conclusion

This study has dealt with the approaches for estimating thenext failure time of offshore plant equipment gas compressorHow to estimate the next failure time based on vibration data

has been proposed by three approaches regression modelbased approach Markov model based approach and theirhybrid approach In theMarkovmodel based approach basedon the gathered vibration state-timestamp data the nextfailure time of a gas compressor has been estimated by thefinite state continuous time Markov model theory In theregressionmodel based approach the linear regressionmodelusing moving average filter has been proposed The hybridapproach takes the advantage of two approaches Dependingon the vibration level one of two approach is applied toestimate the next failure time

To show the usefulness of the proposed approachescase examples and computational experiments based onshaft vibration sensor data were introduced in a case studyAlthough the proposed approaches have some limitations infully evaluating the usability in the real field due to the limitedexamples used in the case study we believe that it will provideoffshore operation companies with a reference for doing theimproved maintenance planning and decreasing equipmentdowntime due to unexpected failures

We can think of several future research works Firstafter suitably tuning the parameters (eg 119899 120578 and regressionparameters) used in our approaches based on lots of field dataour approaches could be applied to not only gas compressorbut also other pieces of equipment the failure of whichcould be being monitored by vibration signals Second thelinear regression model could be improved to the moreelaborate regression model by considering the trade-offsbetween generalizability and goodness of fit In general casesince exponential model seems to be more suitable for thedegradation model of mechanical system it is valuable toconsider more complex regression model rather than thefirst-order linear regression model Third it is possible to


specify the approaches in detail considering the segmentationof ocean environment and equipment operating missionprofile Fourth in this study based on engineers interviewwe assumed that the state transition in the Markov model istime independent However as the future research issue it ismeaningful to consider the time dependency of deteriorationprocess of compressor In addition we did not consider real-time interactive update of the parameters used in theMarkovmodel However since the degradation process is changingover time due to various uncertainties it is also valuable toconsider the real-time update by using the methods suchas MCMC (Monte Carlo Markov Chain) and classic controlmodels Finally other probabilistic methods such as Bayesiannetwork and artificial neural network could be applied todevelop a more elaborated prognostic algorithm

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was financially supported by Korea EvaluationInstitute of Industrial Technology (KEIT Republic of Korea)and the Ministry of Trade Industry and Energy (MOTIERepublic of Korea) through the core technology developmentprogram of Industrial Convergence Technology (10045212predictive maintenance system for the integrated and intel-ligent operation of offshore plant)

References

[1] Y Peng M Dong and M J Zuo ldquoCurrent status of machineprognostics in condition-based maintenance a reviewrdquo Inter-national Journal of AdvancedManufacturing Technology vol 50no 1ndash4 pp 297ndash313 2010

[2] S Telford M Mazhar and I Howard ldquoCondition based main-tenance (CBM) in the oil and gas industry an overview ofmethods and techniquesrdquo in Proceedings of the InternationalConference on Industrial Engineering and Operations Manage-ment pp 1152ndash1159 Kuala Lumpur Malaysia January 2011

[3] A Prajapati J Bechtel and S Ganesan ldquoCondition basedmain-tenance a surveyrdquo Journal of Quality in Maintenance Engineer-ing vol 18 no 4 pp 384ndash400 2012

[4] J Zhu Y Li W Wang et al ldquoOffshore adaptability of the CO2pre-cooling dual nitrogen expander natural gas liquefactionprocessrdquo Advanced Materials Research vol 608-609 pp 1369ndash1374 2013

[5] M Bevilacqua and M Braglia ldquoThe analytic hierarchy processapplied to maintenance strategy selectionrdquo Reliability Engineer-ing and System Safety vol 70 no 1 pp 71ndash83 2000

[6] R Kothamasu S H Huang andWH VerDuin ldquoSystem healthmonitoring and prognosticsmdasha review of current paradigmsand practicesrdquo International Journal of AdvancedManufacturingTechnology vol 28 no 9 pp 1012ndash1024 2006

[7] D T Griffith N C Yoder B Resor J White and J Paque-tte ldquoStructural health and prognostics management for theenhancement of offshore wind turbine operations and mainte-nance strategiesrdquoWindEnergy vol 17 no 11 pp 1737ndash1751 2014

[8] J Lee ldquoTeleservice engineering in manufacturing challengesand opportunitiesrdquo International Journal of Machine Tools ampManufacture vol 38 no 8 pp 901ndash910 1998

[9] L D Lee ldquoUsing wireless technology and the internet for pre-dictive maintenancerdquoHydrocarbon Processing vol 80 no 5 pp77ndash80 2001

[10] LDieulle C Berenguer AGrall andMRoussignol ldquoContinu-ous time predictive maintenance scheduling for a deterioratingsystemrdquo in Proceedings of the IEEE Annual Symposium onReliability and Maintainability pp 150ndash155 Philadelphia PaUSA January 2001

[11] AGrall C Berenguer andLDieulle ldquoA condition-basedmain-tenance policy for stochastically deteriorating systemsrdquo Relia-bility Engineering and System Safety vol 76 no 2 pp 167ndash1802002

[12] C-C Lin and H-Y Tseng ldquoA neural network application forreliability modelling and condition-based predictive mainte-nancerdquo International Journal of Advanced Manufacturing Tech-nology vol 25 no 1-2 pp 174ndash179 2005

[13] W J Moore and A G Starr ldquoAn intelligent maintenance systemfor continuous cost-based prioritisation of maintenance activi-tiesrdquo Computers in Industry vol 57 no 6 pp 595ndash606 2006

[14] S-J Wu N Gebraeel M A Lawley and Y Yih ldquoA neural net-work integrated decision support system for condition-basedoptimal predictive maintenance policyrdquo IEEE Transactions onSystems Man and Cybernetics Part ASystems and Humans vol37 no 2 pp 226ndash236 2007

[15] WWu J Hu and J Zhang ldquoPrognostics ofmachine health con-dition using an improvedARIMA-based predictionmethodrdquo inProceedings of the 2nd IEEE Conference on Industrial Electronicsand Applications (ICIEA rsquo07) pp 1062ndash1067 IEEE HarbinChina May 2007

[16] H M Hashemian and W C Bean ldquoState-of-the-art predictivemaintenance techniquesrdquo IEEE Transactions on Instrumenta-tion and Measurement vol 60 no 10 pp 3480ndash3492 2011

[17] AGruber S Yanovski and I Ben-Gal ldquoCondition-basedmain-tenance via simulation and A targeted bayesian network meta-modelrdquo Quality Engineering vol 25 no 4 pp 370ndash384 2013

[18] J Lee FWuW ZhaoM Ghaffari L Liao andD Siegel ldquoProg-nostics and health management design for rotary machinerysystemsmdashreviews methodology and applicationsrdquo MechanicalSystems and Signal Processing vol 42 no 1-2 pp 314ndash334 2014

[19] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[20] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering Transactions of theASME vol 131 no 3 pp 0310091ndash0310099 2009

[21] X-S Si W Wang C-H Hu and D-H Zhou ldquoRemaining use-ful life estimationmdasha review on the statistical data drivenapproachesrdquo European Journal of Operational Research vol 213no 1 pp 1ndash14 2011

[22] W Wang and H B A Majid ldquoReliability data analysis andmodelling of offshore oil platform plantrdquo Journal of Quality inMaintenance Engineering vol 6 no 4 pp 287ndash295 2000

[23] N Arthur and M Dunn ldquoEffective condition-based main-tenance of reciprocating compressors on an offshore oil andgas installationrdquo in Proceedings of International Conference onCompressors andTheir Systems pp 213ndash221 London UK 2001


[24] P Caselitz and J Giebhardt ldquoAdvanced maintenance and repairfor offshore wind farms using fault prediction techniquesrdquoin Proceedings of the World Wind Energy Conference BerlinGermany 2002

[25] P K Dey S O Ogunlana and S Naksuksakul ldquoRisk-basedmaintenance model for offshore oil and gas pipelines a casestudyrdquo Journal of Quality in Maintenance Engineering vol 10no 3 pp 169ndash183 2004

[26] A G Eleye-Datubo AWall A Saajedi and J Wang ldquoEnablinga powerful marine and offshore decision-support solutionthrough bayesian network techniquerdquo Risk Analysis vol 26 no3 pp 695ndash721 2006

[27] A G Eleye-Datubo AWall and JWang ldquoMarine and offshoresafety assessment by incorporative risk modeling in a fuzzy-Bayesian network of an induced mass assignment paradigmrdquoRisk Analysis vol 28 no 1 pp 95ndash112 2008

[28] E Miguelanez and D Lane ldquoPredictive diagnosis for offshorewind turbines using holistic condition monitoringrdquo in Pro-ceedings of the IEEE OCEANS pp 1ndash7 Seattle Wash USASeptember 2010

[29] HHussinMMuhammad FMHashim and S N Ibrahim ldquoApractical method for analyzing offshore gas compressor systemmaintenance datardquo AIP Conference Proceedings vol 1285 no 1pp 207ndash221 2010

[30] A H de Andrade Melani D W R Silva and G F M SouzaldquoUse of Bayesian network to support risk-based analysis of LNGcarrier loading operationrdquo in Proceedings of the ProbabilisticSafety Assessment and Management (PSAM rsquo14) HonoluluHawaii USA June 2014

[31] S Cho H Jun J Shin and S Choi ldquoA study on estimatingthe next failure time of lng fpso compressorrdquo Korean Journal ofComputational Design and Engineering vol 19 no 3 pp 203ndash213 2014

[32] S Cho H-B Jun J-H Shin H-J Hwang and C Ha ldquoAstudy on the development of prognosis system for offshore plantequipmentrdquo in Proceedings of the 25th International Ocean andPolar Engineering Conference (ISOPE rsquo15) pp 464ndash470 KonaHawaii USA June 2015

[33] B Jones I Jenkinson Z Yang and J Wang ldquoThe use ofBayesian network modelling for maintenance planning in amanufacturing industryrdquo Reliability Engineering and SystemSafety vol 95 no 3 pp 267ndash277 2010

[34] SINTEF Offshore reliability data OREDA participants 2009[35] R K Mobley An Introduction to Predictive Maintenance Else-

vier 2002[36] P W Tse and D P Atherton ldquoPrediction of machine deterio-

ration using vibration based fault trends and recurrent neuralnetworksrdquo Journal of Vibration and Acoustics vol 121 no 3 pp355ndash362 1999

[37] D Anderson ldquoIntroduction to stochastic processes with appli-cations in the biosciencesrdquo Tech Rep University of Wisconsinat Madison 2013

[38] P Goodwin and R Lawton ldquoOn the asymmetry of the symmet-ricMAPErdquo International Journal of Forecasting vol 15 no 4 pp405ndash408 1999

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Algebra

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Stochastic AnalysisInternational Journal of


by enabling network bandwidth data collection and retrievaldata analysis and decision support capabilities for large datasets of time series data [3] Under the new environmentwe can gather the equipment status data related to usageconditions failure maintenance or service events and so onThese data sets enable us to diagnose the degradation statusof the equipment in a more exact way Therefore using thisinformation gives us new challenging issues for improvingthe efficiency of equipment maintenance operations We candiagnose equipment status predict equipment abnormalityand execute proactive maintenance that is doing CBM

Although there have been some relevant research worksso far the CBM is still challenging area Current approacheshave the limitation in detailed methods or validated predic-tive models in particular in the offshore plant domain Thisstudy deals with the approaches that can predict the next fail-ure time of offshore plant equipment (gas compressor) usedin LNG FPSO (Liquefied Natural Gas Floating ProductionStorage and Offloading vessel)

An LNG FPSO is an offshore plant of delivering liquefiedgas from a gas field to customers Recently the demand forLNG FPSO is highly increasing and the demand for LNGFPSO projects will grow along with the increased demandfor natural gas [4] The OampM phase of LNG FPSO requiresheavy charges and more efforts to optimize the cost and toreduce the risks than the construction phase because of itslong life cycle Nowadays due to the fact that an accident ofLNG FPSO in operation causes catastrophic damage manystudies have focused on a maintenance system In this veinthis study focuses on the prognostic approach for the gascompressor equipment in LNG FPSO which is one of themain results for the Korean government supported projectthat is being currently developed since 2013 with the objectiveof implementing the predictive maintenance system for LNGFPSOsThe objective of this study is to develop the algorithmfor estimating the next failure time of a gas compressorbased on gathered vibration sensing and failure data Tothis end in this study finite state Markov model based ap-proach linear regression model based approach and theirhybridmodel have been introduced To evaluate the proposedapproaches the case study and computational experimentsfor compressor equipment have been carried out

The rest of this study is organized as follows First rele-vant previous studies are reviewed and their limitations arediscussed in Section 2 Then Section 3 describes the equip-ment for CBM focused on this study and two approachesfor estimating the next failure time of a compressor andSection 4 introduces relevant case studies and computationalexperiments In Section 5 the hybrid approach combiningtwo approaches is proposed with computational experimentFinally this study is concluded with further research issues inSection 6

2 Literature Review

There are several maintenance policies corrective mainte-nance preventive maintenance opportunistic maintenanceCondition-Based Maintenance and predictive maintenance

Correctivemaintenance is the unplannedmaintenanceHow-ever preventivemaintenance (such as constant intervalmain-tenance age basedmaintenance and imperfectmaintenance)and predictive maintenance (such as RCM (Reliability Cen-tered Maintenance) and CBM) are the types of plannedmaintenance Formore details please refer to Bevilacqua andBraglia [5] or Kothamasu et al [6] Among various mainte-nance policies this study focuses on the CBM

The term CBM is often used with other terms suchas PdM (Predictive Maintenance) PHM (Prognostic andHealth Management) and on-condition maintenance whichcomes from the US Department of Defense and Departmentof Energy In this study we define CBM as a maintenancepolicy which does maintenance action before equipmentfailures happen by assessing equipment condition includingoperating environments and predicting the risk of equip-ment failures in a real-time way based on gathered dataThe benefits of a successful CBM strategy are expected toinclude less regular maintenance the reduction of unsched-uled maintenance and improved supply chain management[7]

Until so far there have been several research works aboutCBM For example Lee [8] introduced the fundamentaltechnologies for remotemaintenance (called teleservice engi-neering) and CBM machine performance assessment andremote diagnosis Lee [9] introduced a new methodology ofCBM called machinery dynamics and data fusion throughremote machinery monitoring He presented an example ofa remote wireless application currently in use for monitoringmachinery in industrial plants Dieulle et al [10] dealt withthe problem related to CBM policy for a single-unit deteri-orating system and proposed the approach to determine theoptimal inspection schedule and replacement threshold withrenewal processes theory Furthermore Grall et al [11] dealtwith a condition-based inspectionreplacement problem fora stochastically and continuously deteriorating single-unitsystem With regenerative and semiregenerative processestheory they tried to find twomaintenance decision variablespreventive replacement threshold and inspection schedulewith the objective of minimizing the long run expectedmaintenance cost per unit time In addition Lin and Tseng[12] combined traditional reliability modelling methods withvibration-based monitoring techniques and artificial neuralnetwork technologies in an integrated system to determinethe health status of machinery namely CMAC-PEM (Cere-bellar Model Articulation Controller neural network-basedmachine Performance Estimation Model) They developeda WPHM (Weibull Proportional Hazards Model) and car-ried out a bearing deterioration experiment to test boththe CMAC-PEM and the WPHM Moore and Starr [13]have reviewed the methods and functionality of criticalityassessments in condition-based monitoring and proposedthe CBC (Cost Based Criticality) algorithm to rank all thealarms arising from conditionmonitoring which could allowoptimized prioritization of maintenance activities Wu etal [14] proposed an intelligent decision support system forthe optimal CBM policy They developed a neural networkmodel that uses bearing vibration information to predict thelife percentage of a machine and the remaining life of the











HP separator

Compressor

Surge tank

Pump

Gas dehydration

Good fluid

Producedwater

Oil

To flare

Gas

Produced water

Overboard

Oil

Export gas

Lift gas














levels




diagram analysis





filtered data


regression model


Hybrid approach

YesNo











Event occurrence


State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax






1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)



1198710119904= infinsum119895=0


(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4



1198710119904 = infinsum119895=0




= infinsum119895=0





= infinsum119895=0












(3)


Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















HP separator

Compressor

Surge tank

Pump

Gas dehydration

Good fluid

Producedwater

Oil

To flare

Gas

Produced water

Overboard

Oil

Export gas

Lift gas














levels




diagram analysis





filtered data


regression model


Hybrid approach

YesNo











Event occurrence


State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax






1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)



1198710119904= infinsum119895=0


(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4



1198710119904 = infinsum119895=0




= infinsum119895=0





= infinsum119895=0












(3)


Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











HP separator

Compressor

Surge tank

Pump

Gas dehydration

Good fluid

Producedwater

Oil

To flare

Gas

Produced water

Overboard

Oil

Export gas

Lift gas














levels




diagram analysis





filtered data


regression model


Hybrid approach

YesNo











Event occurrence


State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax






1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)



1198710119904= infinsum119895=0


(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4



1198710119904 = infinsum119895=0




= infinsum119895=0





= infinsum119895=0












(3)


Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













levels




diagram analysis





filtered data


regression model


Hybrid approach

YesNo











Event occurrence


State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax






1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)



1198710119904= infinsum119895=0


(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4



1198710119904 = infinsum119895=0




= infinsum119895=0





= infinsum119895=0












(3)


Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Event occurrence


State 4

State 3

State 2

State 1

State 0

Time

t101 t110 t201 t112 t123 t134

S (p-

p)m

ax






1198791199041199041015840 = 1205911199041199041015840sum1205911199041199041015840119896=1

1199051198961199041199041015840

(1)



1198710119904= infinsum119895=0


(2)

where 11987100 = 0 and 11987101 = 11205820 1199041015840 = 2 3 4



1198710119904 = infinsum119895=0




= infinsum119895=0





= infinsum119895=0












(3)


Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time




S (p-

p)m

ax

S(p-p)max



[119870119895 sdot (119895 + 1)] = 119889119889119870infinsum119895=0

119870119895+1 = 119889119889119870 ( 1198701 minus 119870)

= 1(1 minus 119870)2

(4)









1120582119904minus1 + 1198710119904minus1

(5)


11205833] +11205823 + 11987103 (6)




11987103 = [12058321205822 sdot 11987112 +11205832] +

11205822 + 11987102 (8)

11987123 = 11987103 minus 11987102 (9)



119871 1199044 = 11987104 minus 1198710119904 (10)












120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120578 = arg min119898

(119898+119903minus1prod119894=119898

V119894) 119899 le 119898 le 119896 minus 119903 + 1 (13)

where




13200radic120587 minus 0) (15)

0 5000 10000 15000 20000 25000 30000

s = 4

s = 3

s = 2

s = 1

s = 0

250

200

150

100

50

0

Time

S (p-

p)m

ax


4 Case Study



1205823 = 12059134sum12059134119896=1

11990511989634 =2015 + 14 + 15 + sdot sdot sdot + 13 = 00738 (16)



11987104 = [0039200738 sdot 11987123 + 100392] + 100738 + 11987103 (17)


11987123 = 11987103 minus 1198710211987103 = [0045300569 sdot 11987112 + 100453] + 100569 + 11987102



1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205833 = 00392

1205832 = 00453

1205831 = 00580

1205823 = 00738

1205822 = 00569

1205821 = 00506

1205820 = 00068

3

2

1

0

4


11987102 = [0058000506 sdot 11987101 + 100580] + 100506 + 1198710111987101 = 100068 = 1479778

(18)




10038161003816100381610038161003816100381610038161003816 sdot 100 (19)











y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y = 07305x minus 20066

yi

2767

6

2768

6

2769

6

2770

6

2771

6

2772

6

2773

6

2774

6

2775

6

2776

6

2777

6

230

220

210

200

190

180

170

160

150

140

Regression line


Time

S (p-

p)m

ax





(20)



5 Hybrid Approach







119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427606 13670327607 13981127608 14309227609 14645827610 14973627611 15311527612 15645527613 15975327614 16306427615 166381 151457 1 127616 169561 154742 1 127617 172655 158027 1 127618 175649 161283 1 127619 178551 164492 1 127620 181348 167653 1 127621 183963 170738 1 127622 186444 173737 1 127623 188752 176637 1 127624 190983 179429 1 127625 193030 182094 1 127626 194859 184623 1 127627 196451 187003 1 127628 197835 189222 1 127629 198999 191266 1 127630 199979 193129 1 127631 200711 194804 1 127632 201280 196288 1 127633 201615 197574 1 127634 201790 198655 1 027635 201793 199531 1 027636 201542 200199 1 027637 201101 200664 1 027638 200427 200924 0 027639 199639 200988 0 027640 198696 200859 0 027641 197652 200553 0 027642 196376 200063 0 027643 194990 199400 0 027644 193359 198557 0 027645 191658 197544 0 027646 189901 196380 0 027647 187956 195065 0 027648 186044 193627 0 027649 184097 192073 0 027650 181980 190401 0 027651 179913 188627 0 027652 177806 186770 0 027653 175629 184834 0 027654 173501 182849 0 027655 171427 180825 0 0

Table 3 Continued

119909119894 119910119894 119910119894 V119894 prod119898+119903minus1119894=119898 V11989427656 169394 178775 0 027657 167483 176727 0 027658 165574 174680 0 027659 163767 172647 0 027660 162018 170651 0 027661 160327 168693 0 027662 158764 166788 0 027663 157409 164966 0 027664 156072 163223 0 027665 154883 161569 0 027666 153795 160009 0 027667 152810 158542 0 027668 151905 157175 0 027669 151295 155928 0 027670 150824 154808 0 027671 150363 153812 0 027672 150066 152942 0 027673 149992 152200 0 027674 149918 151585 0 027675 150113 151108 0 027676 150297 150758 0 027677 150689 150546 1 127678 151243 150480 1 127679 151726 150523 1 127680 152399 150680 1 127681 153234 150968 1 127682 154061 151367 1 127683 154907 151859 1 127684 155768 152444 1 127685 156715 153104 1 127686 157694 153843 1 127687 158639 154638 1 127688 159699 155484 1 127689 160672 156379 1 127690 161683 157307 1 127691 162687 158252 1 127692 163702 159217 1 127693 164565 160182 1 127694 165367 161142 1 127695 166084 162079 1 127696 166677 162977 1 127697 167268 163840 1 127698 167799 164650 1 127699 168114 165395 1 127700 168363 166063 1 127701 168493 166643 1 127702 168592 167132 127703 168541 167530 127704 168372 167830 127705 168135 168035 1






6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














6 Conclusion







Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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