Modeling of Two-dimensional Warranty Policy using Artificial Neural Network (ANN) Approach.

Abstract

Modeling of two-dimensional warranty policy is an important but difficult task due to the uncertainty and instability of data collection. Moreover, conventional numerical methods of modeling a two-dimensional warranty policy involves complex distribution function and cost analysis. Therefore, this paper attempts to present an Artificial Intelligence (AI) technique, which is the Artificial Neural Network (ANN) approach in order to improve the flexibility and effectiveness of the conventional method. The proposed ANN is trained with historical data using multi-layer perceptron (MLP), feed forward back-propagation (BP) learning algorithm. The Logarithmic (logsig) and Hyperbolic Tangent (tansig) sigmoid functions are chosen as transfer function. Four popular training functions are adopted to obtain the best BP algorithm, that are, Levenberg-Marquardt (trainlm), Gradient Descent (traingd), Gradient Descent with momentum (traingdm), and Gradient descent with momentum and adaptive learning (traingdx) back propagation algorithm. This ANN model demonstrated a good statistical performance with the mean square error (MSE) values in this four training function, especially traingd. Finally, the adopted sensitivity analysis has revealed that the proposed model had successfully implemented.

Full text

Modeling of Two-dimensional Warranty Policy
using Artificial Neural Network (ANN) Approach
Hairudin A. Majid, Jun C. Ang, and Azurah A. Samah
Abstract— Modeling of two-dimensional warranty policy is an important but difficult task due to the uncertainty and instability of
data collection. Moreover, conventional numerical methods of modeling a two-dimensional warranty policy involves complex
distribution function and cost analysis. Therefore, this paper attempts to present an Artificial Intelligence (AI) technique, which is
the Artificial Neural Network (ANN) approach in order to improve the flexibility and effectiveness of the conventional method.
The proposed ANN is trained with historical data using multi-layer perceptron (MLP), feed forward back-propagation (BP)
learning algorithm. The Logarithmic (logsig) and Hyperbolic Tangent (tansig) sigmoid functions are chosen as transfer function.
Four popular training functions are adopted to obtain the best BP algorithm, that are, Levenberg-Marquardt (trainlm), Gradient
Descent (traingd), Gradient Descent with momentum (traingdm), and Gradient descent with momentum and adaptive learning
(traingdx) back propagation algorithm. This ANN model demonstrated a good statistical performance with the mean square error
(MSE) values in this four training function, especially traingd. Finally, the adopted sensitivity analysis has revealed that the
proposed model had successfully implemented.
Index Terms— Artificial Intelligence, Artificial Neural Network, Two-dimensional Warranty.
——————————  ——————————
1 INTRODUCTION
two‐dimensionalwarrantyiseitheranimpliedoran
expresscontractbetweenthemanufacturerandcon‐
sumer.Underthiscontract,manufacturersagreetopro‐
videasatisfactoryserviceeitherrepairorreplaceitems
thatfailduringthespecifiedperiodorusage(whichever
comesfirst).Nowadays,consumersalwayscomparethe
productperformance,characteristicsofcomparablemod‐
elsofcompetingbrandsbeforepurchaseaproduct.So,
warrantybecomesamajornewdirectioninmanufactur‐
ingindustrysinceitplaysanimportantroleinproviding
aguidelinetocustomers.
Intheautomobileindustry,accuratepredictionofop‐
timalwarrantyperiodandwarrantycostsisoftensought
bythemanufacturer.LeBlanc[1]mentionedthatitisdif‐
ficulttoquantifytherisksandrewardsofofferingawar‐
ranty.Itisbecausethewarrantyperiodistooshort,as
wellastoolongwhichmaybeunprofitableforthemanu‐
facturers[2].Averyshortwarrantyperiodwillinterfere
withsales,whileaverylongonewillleadtolossesfrom
compensationofconsumerclaims.Hence,applicationof
ArtificialIntelligence(AI)inwarrantymarketismuch
moreinterestingrequisitetoaffirmtherationalityand
accuracyofwarrantypolicyprediction.
Vastresearchefforthasbeendevotedtotheuseof
ANNasapracticalforecastingtool[3]. Accordingto
KhasheiandBijari[4],ANNisoneofthemostaccurate
andwidelyusedforecastingmodelsthathaveenjoyed
fruitfulapplicationsinforecastingsocial,economic,engi‐
neering,foreignexchangeandstockproblems.Apart
fromthat,theusedofhistoricaldatainANNforpredic‐
tionorforecastingisverypopularanditsefficiencyis
provenbymanyresearcherssuchas[5]‐[8].Forexample,
asurveythatwasconductedbyMarzietal.[5]hadused
twenty‐yeardatafromS&P500Europeanindexcallop‐
tionpricestoforecastthefinancialmarket,andXuand
Lim[8]hadusedrawandhistoricaldataintheirstudyto
forecastthenetflowofacarsharingsystem.
Inthispaper,wepresenttheapplicationofanANN
techniquetopredicttheminimumwarrantycostandop‐
timalinspectionintervalduringawarrantyperiod.This
paperstartswithsection1,whichintroducetwo‐
dimensionalwarrantyandresearchproblems.Thisisfol‐
lowedbySection2whichdescribestherelatedworkand
motivationofthisresearch.InSection3,theassumptions
fortheeasinessoftheimplementationoftheresearchand
resultwerepresented.Section4brieflydescribesthe
ANNapproach.Theframeworkisthoroughlydescribed
inSection5andisfollowedbySection6whichdiscussed
theeffectsofANNstructurestowardstheMSEvalues.
Thediscussionofthispaperendswiththeconclusion
whichispresentedinSection7.
————————————————
 H.A.MajidiswithFacultyofComputerScienceandInformation
System,UniversitiTeknologiMalaysia,81310Skudai,Johor,Malaysia.
 J.C.AngiswithFacultyofComputerScienceandInformationSystem,
UniversitiTeknologiMalaysia,81310Skudai,Johor,Malaysia.
 A.A.SamahiswithFacultyofComputerScienceandInformation
System,UniversitiTeknologiMalaysia,81310Skudai,Johor,Malaysia.
A
© 2011 Journal of Computing Press, NY, USA, ISSN 2151-9617
http://sites.google.com/site/journalofcomputing/
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
48

2 RELATED WORK AND MOTIVATION
Acompletehistoricaldataispivotalduringthe
developmentofwarrantypolicy.However,itisalwaysan
exhaustinganddifficultjobfortheautomobileworkshop
tohaveaccurateandcompletehistoricaldata.Historical
warrantyclaimandservicedatausuallycontainpartial
informationasitmayberecordedincorrectlyanddueto
uncertaintyandinstabilityofdatacollection.Hence,
predictionofanaccuratewarrantypolicybecomesavery
difficulttask.ThisviewissupportedbyYang [9]who
statedthatwarrantymodelingiscomplicateddueto
warrantycensoringespeciallyfortwo‐dimensional
warranty.Thesevaguenessrealworldproblemsare
typicallytoocomplexforaformalmathematicalmodel
[10].
Foraconventionalmathematicalmodeling,thefirst
steptopredictawarrantypolicyistomodeltheitems’
failuresandthecostsofrectificationactionsoverthewar‐
rantyperiod[11].Numerousfailureprobabilitymodels
suchasWeibull,ExponentialandLog‐normalhavebeen
developedforautomobilewarrantyclaimsdata[12]‐[14].
Probabilitydistributionisamathematicalfunctionused
tomodelthefrequenciesandprobabilityofoccurrences
overatime.Thosemathematicalmodelinginvolvesever‐
alstages,soitmaytakealongtimetobetrulyproficient.
Thus,inthisresearch,anattemptismadetoapplyanAI
techniqueinwarrantyanalysistoreducetheuncertainty
problemsandspeedupthecomputingtime.
AIapproacheswerebroadlyadoptedinmanyareas
whereconventionalmathematicalmodelwerereplaced
withexpertsystemstoimproveflexibilityandeffective‐
nessofthecorrespondingsystem.However,inwarranty
research,theapplicationofAItechniqueisveryfewsuch
as[15]‐[22].AmongtheAIsubjectarea,softcomputing
techniquewasfoundtobethemostpopulartechnique
thathasbeenintegratedwithwarrantyarea[23]‐[27].The
applicationsofANNinwarrantydomainfromprevious
workarediscussedinvariousresearches[28]‐[30].Hrycej
andGrabert[28]usedfailureprobabilityasgeneralfunc‐
tional.Inparticular,theauthorusedmulti‐layerperceptron
asafunctionalapproximation.Theparametersofmulti‐
layerperceptronweretrainedwithhelpofminimumcross
entropyruleinforecastingthewarrantycostofalterna‐
tivewarrantyconditionscenarios.Inotherwork,Leeat
al.[29]haddesignedanearlyclaimwarningsystemusing
neuralnetworklearning.Precisely,thesystemprotects
bothmanufacturersandconsumersbygiving“prior
warning”aboutabnormalincreaseofclaimsrateatacer‐
tainpointbasedontrendandestimationbymonitoring
variousclaimdata.Leeetal.[30]hadalsosuggested
neuralnetworklearningmodelindeterminingearly
warninggradeofwarrantyclaimsdata,whichincludes
AnalyticHierarchyProcess(AHP)analysisandknow‐
ledgeofqualityexperts.Inyear2008,Leeetal.[23]had
proposedadifferenttoolwhichisappropriateformodel‐
ingatwo‐dimensionalwarrantyplan.
3 ASSUMPTION
Afewassumptionshavebeenmadetosimplifytheim‐
plementationoftheproposedalgorithm.First,theusage
conditionsareassumedtobestatisticallysimilarandthe
warrantyclaimswerereportedimmediately,withnode‐
lay.Second,theproposedANNapproachisconsideredto
besuitableforanymodelandmakeofautomobile.Third,
theinputandtargetedoutputdataoftheANNprocess
areassumedtobecompletelyknown.Fourth,although
thenumberofdatausedinthedevelopmentofANNis
small,itisassumedthatthedataissufficientenoughto
achievetheperformancegoalinthisresearch.
4 ARTIFICIAL NEURAL NETWORK (ANN): AN
INTRODUCTION
ANNorcommonlyneuralnetwork(NN)isanintercon‐
nectedgroupofartificialneuronsthatuseamathematical
orcomputationalmodelforinformationprocessingbased
onaconnectionistapproachtocomputation[31],[32].
AccordingtoPrincipleetal.[33],oneofthemostsignifi‐
cantstrengthofANNisitsabilitytolearnfromalimited
setofexamples.ANNhasbeensuccessfullyusedinsolv‐
ingcomplicatedproblemsindifferentdomainsuchas
patternrecognition,identification,classification,speech,
vision,andcontrolsystems[34].
AnANN,whichimitatesthehumanbraininproblem
solving,iscapableinmodelingthecomplexrelationship
betweeninputandoutputtofindpatternsindata.Typi‐
cally,anANNconsistsofasetofinterconnected
processingelementsornodescalledperceptrons.The
nodesareorganizedindifferentwaystoformanetwork
structurewhereeachANNiscomposedofacollectionof
perceptrongroupedinlayers.Eachperceptronisdesigned
tomimicitsbiologicalcounterpart,theneuronandto
acceptaweightedsetofinputandrespondwithanout‐
put[35].
AsophisticatedANNmayhaveseveralhiddenlayers,
feedbackloopsandtime‐delayelements,whichdesigned
tomakethenetworksaseffectiveaspossibleindiscrimi‐
natingrelevantfeaturesorpatterns[35].Themostwell
knownANNisfeed‐forwardANN.Afeed‐forwardANN
consistsofasetofnonlinearneuronsconnectedtogether,
inwhichtheinformationflowsintheforwarddirection
[31].Amongthefeed‐forwardnetwork,multilayerPer‐
ceptron(MLP)isthemostwidelyandcommonlyused
model‐freeestimators.AMLPconsistsofatleastthree
layers,whicharetheinput,hiddenandoutputlayer.The
inputandoutputlayercontainacollectionofneurons
representinginputandoutputvariables.
TherearethreelearningtypesofANNmodels,which
aresupervised,unsupervisedandreinforcementlearning.
Thenetworkmodifiestheweightbasedonasequenceof
trainingvectorwithanassociatedtargetoutputnode,
knownassupervisedtraining.Ontheotherhand,unsu‐
pervisedtrainingreferstoanetworkthatmodifiesweight
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
49

byassigningthemostsimilarinputvectorstoanoutput
unit.Andthethirdlearningtypeisreinforcementtrain‐
ing,whichliesbetweensupervisedandunsupervised
learning.Amongthevariousneuralnetworkmodels,
backpropagationisthebestgeneralpurposemodeland
isprobablythebestatgeneralization[36],[37].Theback
propagationistheclassicalalgorithmusedforlearning.It
isaniterativegradientdescentalgorithmwhichisde‐
signedtominimizethemeansquarederrorbetweenthe
desiredoutputandthegeneratedoutputforeachinput
pattern[38].Inthisresearch,focusisgiventothefeed‐
forwardandback‐propagationmodelwithmultilayer
perceptron.
5 ANN FRAMEWORK
Fourmainstageswereincludedintheframework.The
stagesaredatacollection,datadesignandpre‐processing,
back‐propagationnetworkdesignandnetworkimple‐
mentation.Figure1showsthemainflowofthisresearch.
5.1 Data Collection
Sevenhundredhistoricaldatasetswerecollectedinthis
studyfromaMalaysiaautomobilecompany,knownas
MalaysianTruckandBus(MTB).Thedatasetscomefrom
thesameautomobileproductknownasHICOMPerkasa.
Eachdatasetscomprisedoftheelementsofdatewhen
theclaimwasmade,dateclaimreceived,enginenumber,
drivingmileageandagewhentheclaimwasmade,pro‐
ductiondateandfailureordefectdata.Thehistoricaldata
isoffive‐yearsperiod,rangingfrom1998to2002.Precice‐
ly,thehistoricaldataconsistofinformationofa100ve‐
hicles.Fromthe100samples,onceamaintenanceservice
isdone,thevehiclestatusandinformationwillberecord‐
edasthehistoricaldata.
AccordingtoGeorgilakisetal.[39],thetaskofdeciding
whichoftheelementstobeselectedasinputvariableis
anarduoustask.Theselectedelementsmustcorrespond
toparameters,whichmeanthatitwilldirectlyorindirect‐
lyaffectthepredictionresult.Inthisresearch,fourmain
inputpatternswereproduced,whicharemileageandage
ofavehicleduringtheservicing,thedefectorfailurerate
forfortycomponentsofavehicle,andthedatarecorded
whichiseitherservicemaintenanceorrepairimmediate‐
ly.Thesehistoricaldataarepassedtothenextstage,the
datapre‐processing.
5.2 Data design and pre-processing
Datapre‐processingordatanormalizationhastobedone
beforeitcanbeusedintrainingthenetwork[40].Accord‐
ingtoBirbiretal.[41]andWang[42],normalizationof
datareferstoaprocessofscalingthenumbersinadata
settoimprovetheaccuracyofthesubsequentnumeric
computations.Theauthorsalsomentionedthatnormali‐
zationhelpsinshapingtheactivationfunctionduringa
trainingprocess.Basedonthisstatement,theelements
withhugedifferentiaamongthedatasuchasmileageand
ageintheinputandoptimalinspectionintervalandmin‐
imumcostinthetargetoutputarenormalizedinto[0,1]
bytheexpression:

  (1)

where
xiisanobservationvalueofthefactori;
xmaxisthemaximumvalueofthefactori;
xministheminimumvalueofthefactori;
Xiisthenormalizedvalueofxi.

Thenormalizeddatawerethendividedinto70groups
(10each)andthetargetoutputsfromeachgroupare
computedusingmathematicalmodel.Finally,thedata
aredividedintothreepartsfortraining,validatingand
testingprocess.

5.3 Back Propagation (BP) Network Design
ThestructureofaBPNetworkdesignincludeselements
asillustratedinFigure2.Eachofthestructureelementsis
thoroughlydiscussedinthissection.


Fig. 2. Structure of Network Design
5.3.1 Network Architecture Determination
Inthestageofanetworkarchitecturedetermination,the
numberoflayersandthenumberofprocessingelements
perlayerareofimportantconsideration[41].Multi‐layer
backpropagationneuralnetworkcompriseofinput,
hiddenandoutputlayer.Itsarchitectureisillustratedin
figure3.Inthisresearch,inputvariablesaretheageand
mileageduringtheservicing,defectorfaultycomponents
Fig. 1. Main flow of this research
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
50

andinformationofmaintenancecheckingorimmediate
repair.Theminimumwarrantycostandoptimalinspec‐
tionintervalwereidentifiedtobetheoutputdatainthe
ANNprocess.


Fig. 3. Structure of back propagation ANN for two-dimensional war-
ranty 
Inthisstage,thesetofdataweregroupedintotwo
setsi.e.inputandoutput.Thedatasetwerearrangedina
matrixformofx(input),andy(output)inacolumn.Fig‐
ure4showsthegenericformofinputandtargetoutput
design.Thedatawerekeyed‐inintoMs.Excelandsaveas
atextfilesothatitcanbeusedinMatlabtools.


(a) Inputdesign(b) Outputdesign
Fig. 4. Input and Output design
5.3.2 Hidden Neuron Number and Transfer Func-
tion Optimization 
Thenumberofhiddenlayerandnodesineachofthe
hiddenlayeraffecttheperformanceofanANN.Ifthe
numberofhiddennodesistoosmall,itisnotenoughto
generalizetherulesoftrainingsample.Otherwise,the
ANNwilltakenoisydataintomemory.Todate,thereis
nospecificmethodtochoosetheoptimalnumberofhid‐
denlayerandthenumberofnodesinhiddenlayer[43].
Fauset[44]introducedaruletodeterminethenumberof
neuronnodesinhiddenlayerasillustratedinTable1.


Amongstthetransferfunctionsusedforhiddenand
outputlayerareLogarithmicSigmoidfunction(logsig)
andHyperbolicTangentSigmoidfunction(tansig).Both
Sigmoidfunctionsareoftenusedinhiddenlayerdueto
theirpowerfulnonlinearapproachcapability[45].Forthe
outputofactivationfunction,therangeoftansigis(‐1,1)
andlogsigis(0,1).Themathematicalequationsforboth
functionsare:

 (2)
 (3)
5.3.3 Training function optimization
Fourpopulartrainingfunctionsusedinthisstudyare
trainlm,traingd,traingdm,traingdx[46]‐[48].
Levenberg‐Marquardtbackpropagationalgorithm
(trainlm)isanetworktrainingfunctionthatupdates
weightandbiasvaluesaccordingtoLevenberg‐
Marquardtoptimization.Itwasdesignedtoapproach
second‐ordertrainingwithouthavingtocomputethe
Hessianmatrix.Trainlmisthefastestbackpropagation
algorithminMatlabtoolbox,andishighlyrecommended
asafirst‐choicesupervisedalgorithm,althoughitdoes
requiremorememorythanotheralgorithm.Thetraining
parametersfortrainlmareepochs,show,goal,time,
min_grad,max_fail,mu,mu_dec,mu_inc,mu_max,
mem_reduc.Thetrainingstatusisdisplayedforeveryshow
iterationofthealgorithm.Thetrainingwillterminatesin
fourconditionsi.e.ifthenumberofiterationsexceeds
epochs,iftheperformancefunctiondropsbelowgoal,ifthe
trainingtimelongerthantimeseconds,orifthemagni‐
tudeofthegradientislessthanmin_grad.Theparameter
muistheinitialvalueforμ.Iftheperformancefunctionis
reducedbyastep,themuvalueismultipliedbymu_dec.
Ontheotherhand,iftheperformanceisincreasedbya
step,thenthemuvalueismultipliedbymu_inc.However,
ifthevalueofmuisbiggerthanmu_max,thealgorithmis
terminated.
TABLE 1
NUMBER OF NEURONS OF HIDDEN LAYER
Formula Proposedby
h=nTangandFishwick
h=n/2Kang
h=2nWong
h=2n+1Lippmann
n=numberofneuronsintheinputlayer
h=numberofneuronsinthehiddenlayer
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
51

Gradientdescentbackpropagationalgorithm(traingd)
isanetworktrainingfunctionthatupdatesweightand
biasvaluesinthedirectionofnegativegradientdescent
ofperformancefunction.Thetrainingparametersfor
traingdareepochs,show,goal,time,min_grad,max_fail,and
lr.Theparameter,learningrate(lr)inTraingdalgorithm,
ismultipliedwiththenegativeofgradienttodetermine
thechangesofweightandbiases.Thelargerthelearning
rate,thebiggerthestep.Ifthevalueoflearningrateistoo
large,thealgorithmbecomesunstable.Otherwise,ifthe
valueoflearningrateistoosmall,thealgorithmtakesa
longtimetoconverge.
Gradientdescentwithmomentumbackpropagation
algorithm(traingdm)allowsthenetworktorespondnot
onlytothelocalgradient,butalsotorecenttrendsinthe
errorsurface.Thetrainingparametersfortraingdmare
epochs,show,goal,time,min_grad,max_fail,lrandmc.The
momentum,mcactslikealowpassfilter,whichallowsthe
networktoignoresmallfeaturesinerrorsurface.Anet‐
workwithoutmomentumcangetstuckinashallowlocal
minimum,whileanetworkwithamomentum,canslide
throughsuchminimum.Theinteractionoflearningrate
andmomentumleadstoanacceleratedlearning[49].
Gradientdescentwithmomentumandadaptivelearn‐
ingbackpropagationalgorithm(traingdx)isanetwork
trainingfunctionthatupdatesweightandbiasesvalues
accordingtogradientdescentmomentumandanadap‐
tivelearningrate.Thetrainingparametersfortraingdxare
epochs,show,goal,time,min_grad,max_failmax_perf_inc,lr,
lr_inc,lr_dec,andmc.Adaptivelearningrate,lr_incand
lr_decattemptstokeepthelearningstepsizeaslargeas
possiblewhilekeepinglearningstable.Ifthenewerror
exceedsthepreviouserrorbymorethanapredefined
ratioi.e..max_perf_incinthenetworkwithmomentum,
thenthenewweightsandbiasesareabandoned.


5.4 ANN Implementation
Thisimplementationstageinvolvesthreeprocesseswhich
arenetworktraining,validatingandtesting.Figure5
showstheflowtodeterminethebestANNmodel.
Fig. 5. The flow to find the best ANN model
5.4.1 ANN Model Development
TheANNdevelopmentprocessstartswithnetworktrain‐
ing,followedbynetworkvalidating.Thetrainingprocess
mayconsumealotoftime.Atthebeginning,thenetwork
modelsaretrainedwithasetofinputandtargetoutput
data.Theparameterssettinginthetrainingfunctionare
changedorderlytofindoutthebestANNmodel.The
networkadjuststheweightcoefficients,whichusually
beginwitharandomset,sothenextiterationwillpro‐
duceaclosermatchbetweenthetargetoutputandactual
outputofANN.Thetrainingmethodtriestominimize
thecurrenterrorsfromallprocessingelements,andthe
globalerroriscalculatedbyaperformanceindex.
Iftheperformanceindexintrainingprocessachieves
thetargetedgoalwhichis0.01thenthevalidatingprocess
willbecontinuedbyanothersetofinputdataandtarget
outputdata.Similartothetrainingprocess,aglobalerror
iscalculatedbyperformanceindex.Thetargetedgoalset
inthevalidationstageisintherangeof0to0.07tobe
accepted.Thenetworkmodelhadundergonetraining
andvalidatingprocesscalleddevelopedsystem.Thede‐
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
52

velopedsystemwithitsweightwillpassedtothenext
processwhichistestingprocess.

5.4.2 Model Performance Evaluation
Thenetworkperformancewasquantifiedbycalculating
MeanSquareError(MSE)forthedifferencebetweenthe
expectedandtargetedoutputdataset.Haganetal.[50]
highlightedthattheperformanceevaluationoftheback
propagationalgorithmformulti‐layerANNisMSE.The
learningalgorithmadjuststhenetworkparametersin
ordertominimizetheMSE.TheexpressionofMSEis:

 (4)
where 
zk isoutputpredicted,
yk isactualtargetoutputand
istheerrorofithoutputnumber1,2,…,n.
5.4.3 Network Testing 
Tes ti ngprocesscanonlystartoffifthetrainingandva‐
lidatingprocesseshadbeencarriedthroughbyachieving
theirgoal.Atthisstep,theneuralnetworkwiththesmal‐
lestMSEvalueisdefinedasthebestneuralnetwork
model.Inthisresearchaninterval(0–0.03)forMSEvalue
oftheoutputaredefinedasthesmallestvalue.
6 EXPERIMENTAL RESULT
AnoptimizedANNstructureisusedtoillustratetheper‐
formanceoftheproposedmodel.Sincetheneuralnet‐
workisanonlinearprocedureandthenetworkparame‐
terswillaffecteachotherthentheadjustmentofeachpa‐
rametertooptimizethewholenetworkisnotaneasytask
[51].ThissectiondiscussestheoptimizedANNstructure
whichproducedtheminimumMSEvalueineachtraining
function.Ananalysisispresentedtoverifytheaccuracy
oftheresult.
Table2and3displaythestructureofthebestANN
modelanditscorrespondingparameterssettingineach
trainingfunctionwhichachievedthesmallestMSEvalue
intestingprocess.
TABLE 2
STRUCTURE OF THE BEST ANN MODEL IN EACH TRAINING
FUNCTION
ItemValue/Character
TraingdTrai ng dmTraingdxTrainlm
Number
ofhidden
layer
ThreeThreeThreeTwo
Input
node
43
nodes43nodes43nodes43
nodes
Output2nodes2nodes2nodes2nodes
nodelogsiglogsiglogsiglogsig
Hidden
node
1x86
nodes
tansig
1x86
nodes
logsig
1x22
nodes
tansig
1x86
nodes
tansig
1x43
nodes
logsig
1x43
nodes
tansig
1x43
nodes
tansig
1x22
nodes
logsig
1x22
nodes
tansig
1x22
nodes
tansig
1x22
nodes
logsig

MSEvalue(goal=0.01)
Training0.01000.01000.01000.0096
Validating0.04240.03690.04040.0580
Tes ti ng0.00980.01720.01940.0113
 
TheBestANNmodelamongthetrainingfunction
TABLE 3
THE PARAMETERS USED IN EACH TRAINING FUNCTION
ParameterValu e
TraingdTrai ng dmTraingdxTrainlm
epochs
300000
(goal
met=
206128)
300000
(goalmet
=106643)
300000
(goal
met=
174933)
300000
(goal
met=
811)
goal0.010.010.010.01
learningrate
(lr)0.30.60.9‐‐
learningdec
(lr_dec)‐‐ ‐‐ 0.7‐‐
learninginc
(lr_inc)‐‐ ‐‐ 1.05‐‐
max_fail5(de‐
fault)
5(de‐
fault)
5(de‐
fault)
5(de‐
fault)
max_perf_inc‐‐ ‐‐ 1.04‐‐
momentum
constant(mc)‐‐ 0.10.9‐‐
InitialMu
(mu)‐‐ ‐‐ ‐‐ 0.006
Mudecrease
factor
(mu_dec)
‐‐
‐‐ ‐‐ 0.1
Muincrease
factor
(mu_inc)
‐‐
‐‐ ‐‐ 10
Maximum
Mu
(mu_max)
‐‐
‐‐ ‐‐ 1.0000e‐
010
min_grad1.0000e‐
010
1.0000e‐
010
1.0000e‐
010
1.0000e‐
010
show100100100100
timeInfinityInfinityInfinityInfinity
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
53

ThebestparameterssettingtoobtainthebestANN
model
6.1 Accuracy Analysis upon Experimental Result
Thederivedoutputandexpectedoutputshouldundergo
de‐normalizationbeforeanalysisprocess.Theoutputsare
de‐normalizedbytheexpression:



where 
X isthede‐normalizedvalueofx
xisthederivedoutputorexpectedoutput
xmaxisthemaximumvaluederivedinnormalization
xministheminimumvaluederivedinnormalization

Thede‐normalizedderivedoutputswhichweregen‐
eratedbythebestANNmodelarecomparedwiththede‐
normalizedexpectedoutputintermsofaccuracyusing
thefollowingequation:

wherezfisthederivedoutputandzeistheexpectedout‐
put.

Table4showsthede‐normalizedoutputandtheaccuracy
oftheproposedANNmodelinderivingtheoptimum
warrantycostandinspectioninterval.
TABLE 4
THE ACCURACY OF THE PROPOSED MODEL

Expected
output
Derived
output
Accuracy
(%)
Inspection
interval
(Year)
0.58330.672384.74
1.41671.268889.56
1.08331.041996.17
War ra nty
Cost
(RM)
178.09169.1895.00
75.6489.1082.21
101.50107.9893.62

Fromthisaccuracyanalysis,itisrevealedthatanaver‐
ageof90percentofaccuracyisachievedbyusingthis
proposedmodel.Itcanbesummarizedthattheproposed
ANNmodelissuccessfullyappliedinderivingtheopti‐
mumwarrantycostandoptimuminspectioninterval.
6.2 Sensitivity Analysis: Statistic T-test
T‐testisahypothesistesttoinvestigatethesignificanceof
twosamplesfromanormallydistributedpopulation.The
T‐testisprobablythebestknowntechniqueandthemost
frequentlyusedstatisticaldataanalysismethodforhypo‐
thesistesting[52]‐[53].
Inthisstudy,T‐testisconductedtoassesstheaccuracy
oftheresultswhichwereobtainedbytheproposedANN
model.Thenullhypothesisusedinthiscasestudyis:


H0: Thereisnosignificancedifferencebetween
thederivedoutput(x1)andtheexpectedout‐
put(x2),thatisx1=x2.
H1: Thereisasignificancedifferencebetweenthe
derivedoutput(x1)andtheexpectedoutput
(x2),thatiseitherx1≠x2,x1<x2,orx1>x2.

Table5and6showtheT‐testprocessfromsteps2to6for
inspectionintervalandwarrantycost,respectively.
TABLE 5
T-TEST FOR THE INSPECTION INTERVAL
Derivedoutput
(x1)
Expectedoutput
(x2)
Replicate10.67230.5833
Replicate21.26881.4167
Replicate31.04191.0833
Σx2.98303.0833
Observation(n)33
Mean( )0.99431.0278
Σd2=Σx2‐((Σ
x)2/n)0.18130.3519
Variance,σ2=Σ
d2/(n‐1)0.09070.1760
pooledstan‐
darddeviation,
σ
0.3651
T‐value0.1122
TABLE 6
T-TEST FOR WARRANTY COST

Derivedoutput
(x1)
Expectedoutput
(x2)
Replicate1169.18178.09
Replicate289.1075.64
Replicate3107.98101.50
Σx366.26355.23
Observation(n)33
Mean( )122.0867118.4100
Σd2=Σx2‐((Σ
x)2/n)3504.90035676.9234
Variance,σ
2=
Σd2/(n‐1)1752.45012838.4617
pooledstan‐
darddeviation,47.9109
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
54

σ
T‐value0.0940
Inthisstudy,95percentofconfidenceintervalisadopted.
Thesignificancelevelandcriticalregionarestatedasbe‐
low.

Significancelevel,α=0.05
CriticalRegion:Z<‐1.96orZ>1.96

BasedonthecalculationsinTable5and6,bothT‐value
forinspectionintervalandwarrantycostlieoutsidethe
criticalregion.Hence,thenullhypothesis,H0isaccepted
at5percentsignificancelevel.Itcanbeconcludedthat
thereisnosignificancedifferencebetweenthederived
outputandtheexpectedoutput.
7 CONCLUSIONS
Thispaperhaspresentedafasterandintelligentwayto
predictminimumwarrantycostandoptimalinspection
intervalduringawarrantyperiodbyusingartificialneur‐
alnetworks.Differentnetworkstructuresweretrained
andvalidatedwithanalyticalresultofmathematical
model.Themodelwasthentestedwithaseriesofhistori‐
caldata.Itwasfoundthatthemostefficientalgorithmfor
modelingthetwo‐dimensionalwarrantypolicyisback‐
propagationlearningalgorithmwithGradientDescent.In
thisresearch,althoughtheamountofexperimentaldata
islimited,significantresultprovesthattheproposedal‐
gorithmiscapabletopredictthetwo‐dimensionalwar‐
rantypolicy.Forfurtherresearch,itisrecommendedthat
otherAItechniquesisusedinmodelingthetwo‐
dimensionalwarrantypolicyinordertoreducethecom‐
plexityandtimeconsumingofconventionalmathemati‐
calmodel.
ACKNOWLEDGEMENT
TheauthorshonorablyappreciateMinistryofScience,
TechnologyandInnovation(MOSTI)forthefundingofE‐
SciencegrantandResearchManagementCenter(RMC),
UniversitiTeknologiMalaysia(UTM)forthesupportin
makingthisprojectasuccess.
REFERENCES
[1] LeBlancB.,“A n a l y s i s ofdecisionsinvolvedinofferingaprod‐
uctwarranty,”AnnualReliabilityandMaintainabilitySymposium,
2008.
[2] ChukovaS.S.,DimltrovB.N.,andRykovV. V. , “Warranty
Analysis(Review),”JournalofMathematicalSciences,vol67,no.
6,1993,34863508,doi:10.1007/BF01096273.
[3] Weigend,A.S.,D.E.Rumelhart,andB.A.Huberman,“Genera‐
lizationbyWei ght‐EliminationwithApplicationtoForecast‐
ing,”AdvancesinNeuralInformationProcessingSystems3
(NIPS*90),1991.
[4] KhasheiM.andBijariM.,“A n artificialneuralnetwork(p,d,q)
modelfortimeseriesforecasting,”ExpertSystemswithApplica‐
tions37(2010),pp.479–489,2009.
[5] MarziH.andMarziE.,“UseofNeuralNetworksinForecasting
FinancialMarket,”IEEEConferenceonSoftComputinginIndus‐
trialApplication,SMCiaʹ08,pp.240‐245,2008.
[6] Tay lo rJ.W.andBuizzaR.,“NeuralNetworkLoadForecasting
WithWe atherEnsemblePredictions,”IEEETran sac tio nOnPow‐
erSystems,vol.17,no.3,2002.
[7] Wan gQ.,YuB.andZhuJ.,“ExtractRulesfromSoftwareQuali‐
tyPredictionModelBasedonNeuralNetwork,”Proceedingof
the16thIEEEInternationalConferenceonToolswithArtificialIntel‐
ligence,ICTAI1004,2004.
[8] XuJ.‐X.andLimJ.S.,“Anewevolutionaryneuralnetworkfor
forecastingnetflowofacarsharingsystem,”IEEECongresson
EvolutionaryComputation,CEC2007,pp.1670‐1676,2007.
[9] YangG.andZaghatiZ.,“Two‐DimensionalReliabilityModel‐
ingFromWarrantyData,”IEEEProceedingAnnualReliabilityand
MaintainabilitySymposium,pp.272‐278,2002.
[10] Kastner,J.K.,“Areviewofexpertsystems,”EuropeanJournalof
OperationalResearch,vol.18,pp.285‐292,1984.
[11] MurthyD.N.P.,andDjamaludinI.,“Newproductwarranty:A
literaturereview,”InternationalJournalofProductionEconomics
79(2002),pp.231–260,2002.
[12] Chattopadhyay,G.andA.Rahman,“Developmentoflifetime
warrantypoliciesandmodelsforestimatingcosts,”Reliability
Engineering&SystemSafety,vol93,no.4,pp.522‐529,2008.
[13] MajeskeK.D.,“Amixturemodelforautomobilewarrantyda‐
ta,”ProceedingsofReliabilityEngineeringandSystemSafety,pp.
71‐77,2003.
[14] AskarK.,DoughertyM.,andRocheT.,“Ag e n t rasedsystem
thatsupportreliabilitytransportengineering,”Proceedingsofthe
InternationalConferenceonApplicationsofAdvancedTech nol og i es 
inTransportationEngineeringinBeijing,2004.
[15] DeepR.,etal.“Bit‐MappingClassifierExpertSystemInWar ‐
rantySelection,”ProceedingsoftheIEEEConferenceonNational
AerospaceandElectronicsinDayton,USA.
[16] Derr,J.H.andR.J.Louch,“Ad v a n c e d methodologyforproject‐
ingfieldrepairratesandmaintenancecostsforvehicleelectron‐
icsystems,”SAE(SocietyofAutomotiveEngineers)Tran sa c ti ons ,
1991.100(Sect2),pp.1‐11,1991.
[17] Hyman,W.A.(1989).“Legalliabilityinthedevelopmentand
useofmedicalexpertsystems.”JournalofClinicalEngineering,
14(2):p.157‐163.
[18] KalerJr,G.M.“Expertsystempredictsservice,”HPACHeating,
Piping,AirConditioning,vol.60,no.11,pp.99‐101,1988.
[19] KasraviK.,“Improvingtheengineeringprocesseswithtext
mining.”ProceedingsoftheConferenceonASMEDesignEngineer‐
ingTechnicalinSaltLakeCity,UT,2004.
[20] Lee,S.andL.M.Chang,“Digitalimageprocessingmethodsfor
assessingbridgepaintingrustdefectsandtheirlimitations,”
Proceedingsofthe2005ASCEInternationalConferenceonCompu‐
tinginCivilEngineeringinCancun,2005.
[21] Lin,P. C . , J.Wan g,andS.S.Chin,“Dynamicoptimizationof
price,warrantylengthandproductionrate,”InternationalJour‐
nalofSystemsScience,vol.40,no.4,pp.411‐420,2009.
[22] LeeS.H.,J.H.Lee,etal.,“AFuzzyReasoningModelofTwo ‐
dimensionalWarrantySystem,”7thInternationalConferenceon
AdvancedLanguageProcessingandWebInformationTe ch n ol ogy ,
Liaoning,PEOPLESRCHINA,IEEEComputerSoc,2008b.
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
55

[23] LeeS.H.,LeeD.S.,etal.,“AFuzzyLogic‐ BasedApproachto
Two‐DimensionalWarrantySystem,”4thInternationalConfe‐
renceonIntelligentComputing,Shanghai,PEOPLESRCHINA,
Springer‐VerlagBerlin,2008c.
[24] Vujosevi eM.,Makajie‐NikolieD.,StrakM.(2004),“FuzzyPetri
netbasedreasoningforthediagnosisofbuscondition,”Seventh
SeminaronNeuralNetworkApplicationsinElectricalEngineering‐
Proceedings,NEUREL2004,Belgrade,2004.
[25] ZhouG.,CaoZ.,MengZ.,andXuQ.,“AGAbasedapproachon
arepairlogisticsnetworkdesignwithM/M/smodel,”Proceed‐
ingsInternationalConferenceonComputationalIntelligenceandSe‐
curity,CIS2008,Suzhou2008.
[26] HotzE.andNakhaeizadehG.,PetzscheB.andSpiegelberger
H.,“WAPS,aDataMiningSupportEnvironmentforthePlan‐
ningofWarrantyandGoodwillCostsintheAutomobileIndus‐
try.”ProceedingsofthefifthACMSIGKDDinternationalconference
onKnowledgediscoveryanddatamining,SanDiego,California,
UnitedStates,pp.417–419,1999b.
[27] Hrycej,T.andM.Grabert,“WarrantyCostForecastBasedon
CarFailureData.”ProceedingsofInternationalJointConferenceon
NeuralNetworks,Orlando,Florida,USA,pp.108‐113,2007.
[28] LeeS.H.,SeoS.C.,etal.,“AStudyonWarning/DetectionDe‐
greeofWar ranty ClaimsDataUsingNeuralNetworkLearn‐
ing.”SixthInternationalConferenceonAdvancedLanguage
ProcessingandWebInformationTe ch n ol ogy (ALPIT),2007.
[29] LeeS.etal.,“Ondeterminationofearlywarninggradebased
onAHPanalysisinwarrantydatabase,”LectureNotesinCom‐
puterScience(includingsubseriesLectureNotesinArtificialIntelli‐
genceandLectureNotesinBioinformatics).2008,Shanghai,pp.84‐
89,2008a.
[30] DomingoM.,AgellN.andParraX.,“Connectionisttechniques
toapproachsustainabilitymodeling,”RevistaInternacionalDe
Tecnologia,SostenibilidadyHumanismo,diciembre2006,no.1,pp.
61‐73,2006.
[31] ZabiriH.andMazukiN.,“ABlack‐BoxApproachinModeling
Val ve Stiction,”InternationalJournalofMathematical,Physicaland
EngineeringSciences4:12010,2010.
[32] Principe,J.,NeuralNetworksandAdaptiveSystems,JohnWiley
andSons:NewYor k,NY,1999.
[33] SozenA,ArcakliogluE.,“SolarpotentialinTurkey,” Applied
Energy,80(1),pp.35–45,2004.
[34] OladokunV.O.,AdebanjoA.T.andCharles‐OwabaO.E.,
“PredictingStudents’AcademicPerformanceusingArtificial
NeuralNetwork:ACaseStudyofanEngineeringCourse,”The
PacificJournalofScienceandTech nol og y ,vol.9,no.1,2008.
[35] LawrenceJ.,IntroductiontoNeuralNetwork:Design,Theoryand
Application,6thed.NevadaCity.CA:CaliforniaScientificSoft‐
ware,1994.
[36] MitchellT.,M.MachineLearning,1stedition,NewYor k: 
McGraw‐HillScience/Engineering/Math,1997.
[37] RumelhartD.E.andMcClellandJ.L.,Paralleldistributed
processing:explorationsinthemicrostructureofcognition,Cam‐
bridge,Mass:MITPress,1986.
[38] GeorgilakisP. S.,HatziargyriouN.D.,DoulamisA.D.,Doula‐
misN.D.andKolliasS.D.,“ANeuralNetworkFrameworkfor
PredictingTra nsfor merCoreLosses,”Proceedingsofthe21st
1999IEEEInternationalConferenceonPowerIndustryComputer
Applications,PICAʹ99,pp.301‐308,1999.
[39] NorHaizanMohamedRadzi,HabibollahHaron,andTuan 
IrdawatiTuanJohari,“LotSizingUsingNeuralNetworkAp‐
proach,”Proceedingofthe2ndIMT‐GTRegionalConferenceonMa‐
thematics,StatisticsandApplications,UniversitiSainsMalaysia,
Penang,pp.13‐15,June2006.
[40] BirbirY., NogayH.S.andTop uzV., “EstimationofTotal Har‐
monicDistortioninShortChordedInductionMotorsUsingAr‐
tificialNeuralNetwork,”Proceedingsofthe6thWSEASInterna‐
tionalConferenceonApplicationsofElectricalEngineering,Istan‐
bul,Turkey,pp.206‐210,2007.
[41] Wang,S.,NeuralNetworkApproachtoGeneratingtheLearning
Curve,INFOR.31(3),pp.136–150,1993.
[42] GaoM.,SunF.,ZhouS.,ShiY.,ZhaoY.andWangN.,“Perfor‐
mancepredictionofwetcoolingtowerusingartificialneural
networkundercross‐windconditions,”InternationalJournalof
ThermalSciences48(2009),pp.583‐589,2009.
[43] FausettL.V.,FundamentalsofNeuralNetwork:Architecture,Algo‐
rithms,andApplications,N.J.:Prentice‐Hall,1994.
[44] ZhangY., LiW.,ZengG.M.,TangL.,FengC.L.,HuangD.L.
andLiY.P. , “NovelNeuralNetwork‐BasedPredictionModel
forQuantifyingHydroquinoneinCompostwithBiosensor
Measurements,”EnvironmentalEngineeringScience,vol.26,no.
6,pp.1063‐1070,2009.
[45] MenH.,LiX.,WangJ.,andGaoJ.,“AppliesofNeuralNetwork
toidentifygasesbasedonelectronicnose,”IEEEInternational
ConferenceonControlandAutomationFrC4‐2Guangzhou,CHINA,
2007.
[46] SirajFadzilah,Yus of fNoorainiandKeeL.C.,“Emotionclassifi‐
cationusingneuralnetwork,”InternationalConferenceonCompu‐
ting&Informatics,ICOCIʹ06,pp.1‐7,6‐8June2006.
[47] ZhouM.,ZhangS.,WenJ.andWangX.,“ResearchonCVT
FaultDiagnosisSystemBasedonArtificialNeuralNetwork,”
IEEEVehi cl ePowerandPropulsionConference(VPPC),Harbin,
China,2008.
[48] KeJ.,LiuX.,andWan gG.,“TheoreticalandEmpiricalAnalysis
oftheLearningRateandMomentumFactorinNeuralNetwork
ModelingforStockPrediction,”AdvancesinComputationandIn‐
telligence.LectureNotesinComputerScience,vol.5370/2008,pp.
697‐706,2008,doi:10.1007/978‐3‐540‐92137‐0_76.
[49] HaganM.T., DemuthH.B.andBealeM.H.,NeuralNetwork
Design,PWSPublishingCompany,1996.
[50] LeeT.L.Neuralnetworkpredictionofastormsurge.OceanEngi‐
neering33,pp.483–494,2006.
[51] Marryanna,L.,SitiAisahS.andSaifulIskandarK.,“Water
qualityresponsetoclearfellingtreesforforestplantationestab‐
lishmentatBukitTerek F.R.,Selangor,”JournalofPhysical
Science,18(1),pp.33‐45,2007.
[52] NeideenT.andBraselK.,“UnderstandingStatisticalTests,”
JournalofSurgicalEducation,pp.93‐96,2007.
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
56

Hairudin Abdul Majid has received Diploma and Bachelor of
Science in Computer Science-majoring Industrial Computing from
Universiti Teknologi Malaysia in 1993 and 1995 respectively. In
1998, he obtained his M.Sc. in Operational Research and Applied
Statistic from University of Salford, UK. His PhD thesis in Warranty
and Maintenance(Submited). Currently, he is a lecturer in Faculty of
Computer Science and Information System, Universiti Teknologi
Malaysia. His research interests focused on Image Processing, Op-
erations Management and Warranty and Maintenance. Mr. Hairudin
received Excellent Service Award by Universiti Teknologi Malaysia in
2004 and Excellent Staff Award by ISS Service in Manchester UK in
2006. Mr. Hairudin is the author of about 19 papers, 1 book chapter
entitled ‘Recent Operations Research Modelling and Applications
(Warranty Modelling)’ (UTM, 2009) and 1 text book entitled ‘Permo-
delan Simulasi’ (UTM, 2000). He has been a member of UK Opera-
tional Research Society and an active member of Operations and
Business Intelligence (OBI) Research Group.
Ang Jun Chin obtained her M.Sc. in Computer Science from Un-
iversiti Teknologi Malaysia in 2011 and Bachelor of Computer
Science in 2009. She is currently working as a system analyst in
Singapore.
Azurah A Samah has received the Diploma and Bachelor from Un-
iversiti Teknologi Malaysia in 1991and 1993 respectively. In 1996,
she obtained her M.Sc. from the University of Southampton, UK and
recently in 2010, she received her Ph.D from Salford university, UK.
Currently she is a lecturer in Faculty of Computer Science and In-
formation System, Universiti Teknologi Malaysia. Her research inter-
ests encompass Image Processing, Soft Computing Techniques and
Operational and Simulation Modeling.
JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617
https://sites.google.com/site/journalofcomputing
WWW.JOURNALOFCOMPUTING.ORG
57