voLUME 5, I\UMBER 2
DECEMBER 2OO9
rssN 1693-5098
JOURNAL OF
METH
||IIS
|IUA]IIITAIIUE
JOT]RNAL DEVOTED TO TIIE MATHEMATICAL
AND SIATISTICAL APPLICATIONS IN VARIOUS
FIELDS
QUANTITATIVB METHODS
QUANTITATIVE METHODS provide a forum for communicationbetween statisticiansand
with the usersof appliedof statisticaland mathematicaltechniquesamonga wide
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range of areas such as operationsresearch,computing,actuary,engineering,physics, biology,
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on boththe relevanceof numericaltechniquesand quantitative
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ideas in applied areasand theoreticaldevelopment,which clearly demonstratesignificant,applied
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Editor in Chief:
Dr. Noor Akma Ibrahim
'1ii#lJl'.1'ii:J$?,HrJifr
,,,,?1?ffffii;"j#ffiffi
'.*"
Editorial Assistant:
Dr. JamalI Daoud
Departmentof Sciences,
Facultyof Engineering
InternationalIslamicUniversitvMalavsia
Editors:
Dr.ZhenbngZeng
ShanghaiInstitutefor BiologicalSciences
ChineseAcademyof Science
Dr.FaizA.M. Elfaki
Departrnent
of Sciences
IntemationalIslamicUniversity,Malaysia
Dr.RaihanOthman
Departmentof Science
InternationalIslamicUniversity,Malaysia
Dr. Mat RofaIsmail
Department
of Mathematics
UniversityPutraMalaysia,Malaysia
Dr. IsmailBin Mohd
Departmentof Mathematics
UniversityMalaysiaTrengganu,
Malaysia
Dr.MaherTaka
Facultyof PlanningandManagement
Al Balqa, AppliedUniversity,Al-Salt,Jordan
Chen-juLin, Ph.D
Dr. Hon Wai Leong
Dept.of IndustrialEngineering&Management
Departmentof Mathematics
Singapore
YuanZe University,Taiwan
NationalUniversityof Singapore,
Dr. AndreasDress
Dr.MustofaUsman
Max PlankInstituteor Math.in the Sciences Departmentof Mathematics
22-26,D04103Leipzig,Germany Universityof Lampung,Indonesia
Inselstrasse
Dr.PieterN.Juho
Departmentof Mathematics,
NatalUniversity,SouthAfrica
Dr.Muhammad
AzamKhan
Universityof ScienceandTechnologyBannu,
NWFP-Pakistan
Dr. SannayMohamad
Departmentof Mathematics
UniversityBruneiDarussalam,
BruneiDarussalam
Dr.ZainodinHaji Jubok
Centrefor AcademicAdvancement
UniversityMalaysiaSabah,
Malaysia
Vol. 5, No.2 December2009
QUANTITATIVE METHODS
ISSN 1693-5098
TABLE OF CONTENTS
Data EnvelopmentAnalysisFor StocksSelectionOn BursaMalaysia...
OngPoayLing,AntonAbdulbasah
Karnil,MohdNainHj. Awang
On Group Method Of Data Handling(GIDID ForData Mining.......
Iing Lukman,Noor Akma lbrahim,IndahLia PuspitaandEka Sariningsih
26
StemBiomassEstimation of RoystoneaRegiaUsingMultiple Regression
NorainiAbdullah,ZainodinHj.Jubok,AmranAhmed.
39
ContinuedFractionand DiophantineEquation..........
Yusuf Hartono
49
Risk Analysisof Traffic And RevenueForecastsFor Toll RoadInvestment
In Indonesia.................
I RudyHermawanK., Ade Sjafruddin,danWekaIndraDharmawan
54
UseOf RanksFor TestingFixedTreatmentEffectsIn BasicLatin SquareDesign............
Sigit Nugroho
61
Analysisof Goodcorporate Governancestructure Effectson Bond Rating.......
EindeEvana,AgriantiKSA, andR. WeddieAndryanto
69
A New Method for GeneratingFuzry Rulesfrom Training Data and Its ApplicationTo
ForecastingInflation Rateand InterestRateof Bank fndonesiaCertificate.
AgusMamanAbadi,Subanar,
WidodoandSamsubar
Saleh
78
Mean-var Portfolio OptimizationUnder Capm with Lagged,Non ConstantVolatilty
andrhe LongMemory
Erfect
....i;k;;;ffi;;;;.ilffi;ilr;dt
84
I"
JOURNAL OF QUANTITATIVE METHODS
Vol. 5, No. 2, Dec€mber2009
decq
there
rssN 1693-5098
accut
90t!!d
Indoo
Wang
A IYEW METHOD FOR GEI\TERATING WZZ,V RTJLESFROM TRAINING
DATA
AND ITS APPLICA.TION TO FORECASTING II\TFLATION RATE AIYI} INTEREST
RATE OF BAhiK INilX}IYESIA CERTIFICATE
The n
fiEr
rule h
intfi!
rate N
AgusMamanAbadir,subapaf, widodq?andslmsubarsarehr
'Departnentof Mathematics
Fscultyof MathemsticsandNahnalSciences,
YogyakataStateLJniversity,
Indonesia
2.w|
2Departrnentof
ffiffi13:r:ffiffi ffiffi *t*rii"o'ulsciences.
In thir
GedjahMadsUniveaslty,Indonesia
3Department
of,.*"ff*rYffi
tdl#ffi
Suppo
ifrfi u*"^ity,
rndonesia
":Hrffitr;*j;$ff
yoryakarh 552g1,Iodonesia
JL Humaniora"
Bulaksumur
r , , €[
by thc
Step L
For er
2subanar@vahoo.conr.
2widodo-math@vahoo,conr.
Email: rmamanabadi@.vmail.cornshumas(opaue.uern.ac,id
(Received l0 April 2@9, accepted 20 Octobsr 2009, published
Decernber20@)
Abstract' Table tookup schemeis a simple method to construct fuzzy rules at
fuzzy model. That can be usedto
overcome the conflicting
by determining each rule degree. The weakness of'furzy model based
ryle
on table
lookup schemeis that the fuzzy rules may not be complete so the fuzzy rules
can not cover all values in the
domain' In this paper a new method to generatefuzzy rules from traininj oata wiil
k proposed.In this method,
all completefuzzy rules are identified by firing strengthof each possibie fuzzy
rule."rhen, the resulgedfurzy
rules are.usedto dosignfuzzy mo&l- Applicati,onsoittr" p.opor"o methodto predict
the Indonesianinflation
rale and interestrate of Bank Indonesiacertificate
{BIC) wilf be discussed.Thb predictioasof the Indonesian
inflation rate and interest rate of BIC using the proposed method have
a higher accuracy than those using the
table lookup scheme.
Keywords: fi'uzy rule, fuzzy model, firing strength of rule, inflation rate, interest
rate of BIC.
l. Introduetion
Designingfuz4 rule baseis one some stepsin fuzzy modeling. Fuzzyrule
baseis the heart of ruzzymodel.
,of
Recently,fuzzy model was developedby some researchers.waig, L.x. (1997)
createdfuzzy model basedon
table lookup scheme,gradient descenthaining, recursive least squires and clustering
methods. The weaknessof
the fuzzy model basedon table tookup schemeis that the fuz4 rulebase may not
b€ complete so the fuzzy rule
-S.M.
can not cover all values in the domain' wu, T-P, and Cben,
(ts9S) &sigred mem'terstripfunctions and
fuzzy rules from training data using a -cut of fuzzy sets.In wu and Chen's .Ltto4
determiningmembership
functions and fvzzy rules.needs large cromputations.To reduce the complexity of
computation, iu*, y., et at
(1999) built a methodto decreasefuzzy rules using singularvalue decomposition.
fen, J', et al (1998) developed fuzzy model by combining globat and local learning to improve the
interprelability. New form of fuzry inference systemsthat was h-igily interpretable based
on-a ludlcious choice
of membershipfunction was presentedby Bikdash, M. (1999). C".."., LJ., et al (2005)
constructedTakagiSugeno-Kangmodels having high interpretability using Taylor seriesapproximation.Thin, pomares,
H., et al
(2002) identified structureof fuzzy systemsbasedon
rules thai can decidewhich inpui uariattes .-nust
"o*pi*"membenhip functions
be taken into accountin the fuzzy systemand how many
are neededin every selected
input variable-Abadi' A.M., et al (200sa) designedcompletefuzzy rutes using singular
value decomposition.In
this method,the predictionenor of training datadependedonly on taking the n-umdr
of singularnaiues.
Fuzzy models have been applled to many fields such as in communications,economics,
engineering medicine,
etc. Sp€cially in economics, Abadi, A.M., et al (2006) showedthat forecastingthe lndoneJian
inflat*ion*t" by:
fuzzy model resultsdmore accuracythan that by regressionmcthod. Then,Abii, A.M., et al (2007)
constucted
fuzzy time seriesmodel using tablc lookup schemeto forecastthe interestrate of Bank
Indonesiacertificate
(BIC) and its result gave high accuraoy.Then, Abadi, A.M., et al (2008b)
showedthat forecastingIndonesian
inflation rate basedon fuzzy timc series data using combination or talte lookup scheme
and singular value
Simih
Stcp 2.
For a
sets 4
variaUr
determ
!*0,
Stcp 3.
From sl
possiHr
but diff
degree
pair(4,
Step 4. {
The nrk
with anl
from hu
Stcp 5. (
We can
fivzy m
defined I
much les
that the I
fuzzf'rd
3.
Ner
Given d
/ , ef a, ,
the follor
79
Jownal of Quan|itdive Metho&, Va5 No'2, December 2009, 7833
Based on the previous research'
decompcition methods had a higher accufttcy than that using Wang's methodthat give good prediction
rules
fuuy
in
determining
especially
model
there are irreresting topics in frzzy
in tbis paper, we design
accuracy. To oveircomethe weakness of tabll bokup scheme (Wang's method),
;*ui;
is uspdto predictthg
n rw *rer tI {wq *oa"l *ing-ruing strengthof rule Ta.thu! its rcsult
prediction accuracy than
higher
Indonesian inflation rate andinterest rat€ ;f BIC. The pmposed method has
and
interest rate of BIC'
rate
inflation
lndonesian
the
predicting
to
W*g', method in its applications
Wang's me{hod to coostruct
The rest of this pper is organized as follorrs. In section 2, we briefly review the
using.filng strenglh of
rules
fuzzy
complete
to
constmct
method
new
pr65ent
a
*oa.f. In'xfoon 3, we
n
'"y bssed ur training OaL fn section 4, we apply tlre proposed_'n-.9d to forecasting the inflation rate and
rule
in the forecasting inflation
inter€st rate of BIC. We also comparethe propoied mettroOwittr the Wang's method
q1p
in
section
5'
dispussod
rate and intp.rest1-3tpof BIC, FiqAlly, Someconclusions
2. Wang's method fordesigning
fuzzy rules
from wang, L.X. (1997).
In this sestior,"we will introduce the wang's method to consEuct fuzzy rules referred
where
p=loZ'3,"',il
(x,o,x2r,--,x,ri!o),
dulai
input<nrtput
l{
Supposetlrat we are given the following
is
Wang'smethod given
x,, efa,, F,lc R arrd % e1ur, FrlcR , i: 1, 2, "', n'Designingfuzzy modelusing
by the folloudng stePs:
Siep l, Oefins &zzJ setsto gorer the input and output domains'
l ,i ,i :l r2,...,N i w hi charecompl ete i n fa , , p, l'
F or e a c h s p a c[a
e ,, g ,l ,i :1 ,2 ,...,n ,d e fi ne N ,fi tzrzysets
= l,2o .-.,Nv which are complek in lar, f ,7'
Similarly, define N, fuzy ss{s Br ,i
Stcp 2. Generateone rule from one input-output pair'
I : 1,2' "', n in fiz4
For eachinput-ou9utpair(r,r,rrr,. ..,x,.;!r). determinethe membershipvalueof x,r,
= 1,2' "', N"' Then, for eachinput
sets 4 , j = l,2, ..., Ntandmembershipvalue of yrin fuzzy sets Bi ,i
Indherwor4
variabler,r, i=l,2,...,n,determinethe fuzzysetinwhichr,, hasthelargcsmembershipvalue.
'i =lo?,-.N,.
Similarly, ddermine Bn such that
such that tto(t,)2lt,(",),
determine t
pd,(Jr)>lto(t),1=1,2,"-Nr'Finally,wcconstructafrvzyIF-THENrule:
( l)
IF 4 i s 4 a n d x,i s.l ' and' .-andr,i sl i ,TH E N yi sftr
Stcp 3. Compute degreeof each rule dqsigred in step2'
is large, then it is
frol step 2,'one rui-eis gener*ed by onJinput-ouput pair. If the number of input-output-data
have
same IF parts
rules
if
the
rules
possible itrat there ur" tti" conflicting rules. Two rules be.comeconflicting
in
step 2' The
rule
designed
each
tro
a
degree
we
assigt
pnoblem,
To resolie this
but different THEN parts.
'defined
input-output
the
by
(l)
is
consuucted
the
rules
s-uppase
follows:as
degree of rule is
its degrreis definedas
pair(x,o,xr",..-x,rilr)
'then
= tt G,o)an (t,)
D(rute)
l] o
Stcp 4, Conshuct the fuzzy rule base.
2 that do not conflict
The rule baseconsists otitre following thre€ sets of rules: (1) The rules designedin Step
(3) Linguistic rules
d"se";
maximum
group
has
the
that
a
conflicting
froir
with any other rules; tii rrt"
-r"
from human exPerts.
Step 5. Consnuct the fuzzy model using thefiuzy rule base'
t*ty inferelnceengine and defuzzifiet combined with the fuzzy rule base to desigrr
We can use any nofi"ri
of the fuzzy sets
fuzzy model. ti ttt" nurnU"t of raining data I N and the number of all possible combinations
method may be
defined for the input variables is l-t N, , then the number of fuzzy rules generatedby Wang's
. Then, the fuzzy rule basegeneratedby this method may not be complete so
and fl{
^/
we will design the
that the fuzzy rules can not cover all values in the input spaces.To overcome this weakness,
fu2ry des coveringall valuesin input sipaces'
much less than both
3. Newmethodfor constructingfury rules
cRand
Given the following N training datlt (x,o,xrr,-'.x,0;!o), p=1,2,3,"',N where x,uela,'p,l
is givenby
y, eld,, 0]c n, i: 1,2,..., r. we will inhoducea new methodto designfuzzydes. The method
the following stePs:
Jownal of euantitative Metho*, vo.s No.2,Deceuber 2009, zg{.3
g0
Step 1. Dsfine fuzzy setsto cover the input and output space&
For eachspace[4,, f ,1, i = 1,2, ..., r, defrne N, fiEq *ts A! j = 1,2,..., /{ which arecomplete
,
andnormal
infa,,f ,1. Similarly,definet{, fuzzysetsBr,j=1,2,...,N, whicharecompleteandnormal
infa,,frl.
Step 2. Determine al! po.ssible antecedentsof fuqy rulg candidates.
Based on the Step I' there ut" nlf, antecedents of fuz4
wherr
(4).o
(5). 0
defrrz
defuz
rule candidates. The antecedent has form
*
"4 is 4n and x. is 4 nd... and x" is {' " simplified by
4 andA! nd ...nd 4 ". For example,if we have
two inpul and wp deline two fuzzy scts A, A. for first input spoee4nd f,,,
C, for second inpql spaee"then all
possibleantecedcnts
of fitzy rule candidates
ue A,andC,;.1 atdCr 4*dC,;
\wrdCr.
Step 3. Determine consequenceof each fuzzy rule candidate.
For each antecedent4 nd At and... andA!, the consequenceof fuzzy rule is determined by firing
strengthof
the rule uo&,)u*Qr)...P+Q,,)ttn(l)based
on the trainingdata-Choosingthe consequence
is done as
follows:
For any Eaining data(xrr,xr,,...,x"ri-)rr)andfor any tuzzy setgr, choosegr such
that
It*(x'e)lt*(x,r)...F4(x,".)tr",(ln)2
(4o.,rr,,,...,x,,.;yo.).
uo(\,)tro(x,,)...!t+(x^")ltn(l),forsome
If thereare at leasttu'oBr suchthat uo(xrn\,.pe(x,,.)tturo)
> p4!,(q)...rt*(x",)rtu,(%),
thenchoose
one of 8'' . From this step, we have the fuzzy rule:
iF -r, is A( andx, is A{ and... and x. is lj. , THEN y is B"
so if we continue this step for every antecedent,we get n N, complete fuzzy rules.
Table t
method
constr[l
tudfg
fuzzifiG
Step 4. Construct fuzzy rule base.
The fu:zy rule baseis constructedby the fl l', fuzzy rules designedby Step 3.
S*p 5. Design fuzzy model using fuzzy rule hse.
Fulv model is designed by combining the fuzzy rule base and any fuzzifier, fuzzy inference
engine and
defuzzifier- For example, if we use singleton fuzzifier, produc,tinference engine and ont", uu"nag"
defuzzifier,
then the fuzzy modcl has form:
M ,
Lb,llp.,(x,)
"j='
f (x,xr,...,x,)=
'n
Lp,ttr@,)
rvhereM is the numberof rules.
lrom Step 3, the sel of fuzzy rules constructedby this me&od contains fuzzy rules designed
by the Wang's
method.'Ihereforethe proposedmethodis the generalizationof the wang's method.
4. Applicationsof the proposedmethod
In this section' we apply the proposedmethod to formast the Indorresianinflation rate and inter€st rate of BIC.
The proposedmethodis implementedusing Matlab 6.5.1.
4.1 Forecasting inflation rate
The data ofinflation rate arelaken from January 1999 to January2003. The data fiom January 1999
to March
2002 are usedto training and_thedata from Aprit 2002 to January2003 are usedto testing. ihe procedureto
forecastinginflation rate basedon the proposedmethodis given byihe following steps:
(l)' Define the universeof discourseof each input. In this paper,we will predict the inflation
rate of month t
using inflation rate data of months &-t and &-2 so we will builA fuzzy model using 2-input and l-output. The
universoofdiscourse ofeach input is definedas [-2, 4].
(2). Define fuzzy setson universeof discourseof each input such that fuzzy setscan cover the input
spaces.we
dsfine thirteenfuzzy sets 4,4,...,1\, that are normal and completeon
[-2, 4l of eachinput spacewith Gaussian
membershipfunction.
(3). Determine all possible antecedentsof fuzz1 rule candidates.There are 169 antecedents fuzzy
of
rule
candidatesand the consequ€nce
ofeach antecedentis chosenusing Step 3 ofthe proposedmethod.So we have
169 fu24 rules in the form:
R,: "lf -tr_2 is
4
*d .r*-, is 1b, tUeN xo is AJ.
Table t
in{bneo
infereoa
on the I
Figure I
gl
IN
Josnal of gnntitdive Metho&, Yo.S No.2,December2h|g,Z8A3
gl
wherel:1,2, ,.."169,it,iz=1,2,.,.,13
andAJ e14,4,-.,4r1 .
(4). Constructfuzy rulebasefrun frrzzyrulesdesignedin St€p3.
(5). Oesigrrfu2ry model combiningthe fuzy rule basc and cartainfizzifier, fiuzy inferenceengineand
defuzafier.In this paper,we usesingletonfuzdfrer, minimumand produstinfererceengines,centeraverage
defuzzifier.
IE
Teble l. Comparison
of mean$quare€frorsof f-orpeasting
inflationrate
usingthe Wang'smethodandproposedmethod
!rt
rd[
Methods
Numberof
fuzzyrules
Wang's
method
Proposed
method
23
169
0.43436
lor
B !5
h
"r,
le
f
k-
MSE of training data
Pmduct
inference
eneine
Minimum
inference
ensine
0.47t60
MSE of testing data
Product
inference
ensine
0.45780
Minimum
inference
ensine
0.35309
0.35286
0.407E7
o26097
0.26734
Table I shows a comparison of the MSE of training and testing data basedon the Wang's me{hod and proposed
method. There are twenty three fuzzy rules generated by the Wang's method and there are 169 fuzy-mles
constructed by the proposed rnethod, The predistion of inllation rat€ by the Wang's method with singl*on
fuzzifier, product inference engine and center averagedefirzzifier has a higher accuracythan that with singleton
fitnifier, minimum inference engine and c€nt€r averagedefiuzifier.
Table I illushates that the forecasting inflation raie by the proposedmethod with singleton fiszzifrer, minimum
inference engine and center averagedcfuzzifier has a higher accuracythan t$at with singleton fv.ifier, product
inference engine ald oent€r averagedefuzzifier. The true values and prediction values ofthe inflation rate based
on the Wang's method and the proposed method using minimum and product inference engines are shown in
Figure I and Figure 2 resp@ively,
f5
x
Ftg. l. The prediction and true values ofinflation rate basedon
the Wang's method using: (a) minimum inference engine;
(b) product inference engine
bcrr
3r f ,
ai
Tht
.l c
lm.
NE
brc
Fig. 2. The predictionand fue valuesof inflation rate based on
the proposedmethodusing:(a) minimum inferenceengine;
(b) product inference engine
Journal of Quantitative Methods, Vo.S No.2,December2009, 7843
tz
4.2 Forecasting interest rate of Bank Indonesia certificate
5. Cr
The data of the intercst rate of BIC fiom January 1999 to December?00! afe us€d to Eerning flld th9 data from
January2002to February2003 are usedto lesting.The interestrate of BIC of P monthrvill be predictedby data
of interestrate of BIC of (t-l)* and (t-2)h months so we will build fuzzy model using 2-input and l-output
which universeofdiscourse ofeach input is [10,401. Sevenfuzzy sets are de{ined in dvery input spacewith
Gaussianmembershipfunction. The olher st€psto forecasting the interest rate of BIC Usethq same stepsof the
predicting inflation rate.
In thb
metiod
valuesi
d"sipc
We ap
for€cal
than ft
input r
Teblc 2. Comparisonof mean squar€errors of predicting interest rate of BIC
using thg Wang'$ melhod and proposgdmethod
Method
Wang's
method
hoposed
method
Number
of fuzzy
rules
MSE of trainins data
MSE of testine dafa
Minirnum Product Minimum Product
inference inference inference inference
engine
engine
engine
engine
12
0.98759
49
0.9536r
otrimd
optirnd
Average forecasting
errors oftesting data
6. Ach
(o/o\
Theau
thisre*
Minimum
int'erence
ensine
Product
inference
ensine
0.46438 0.55075
3.8568
4.4393
Refera
0.91682 o.2E3Z:7 0.24098
2.q973
2.7813
lll Ah
1.0687
Table 2 showsthat there are twelve fuzzy rules generatedby Wang's method and there are forty nine fuzzy nrles
constructedby proposed method. The alerage forecasting erors of the prediction of the interest rate of BIC by
Wang's methodand proposedmethodare 3.8568Voand2.78l3 o/orespectively.
qt
t2l Ah
beJ
SFJ
t3l Ah
unr
,1p
[4] Ah
t ol
siq
[5] Bft
!u
[6] Hcn
427
lzl Pq
fitza
l8l \r"r
[9] \r'rr
Fig. 3. the predictionand true valuesof interestrete of BIC basedon the Wang's method
using:(a) minimum inferenceengine; (b) productinferenceengine
From Table 2, forecastingthe interestrate of BIC basedon the Wang's methodusing minimurn inferenceengine
has a good accuracy. Based on the proposed metho4 forecasting of the interest rate of BIC using product
inference engine has a good accuracy. The MSE values and average forecasting errors based on the proposed
method are smaller than those basedon the Wang's method, Figure 3 and Figure 4 show the true valuesand
predictionvaluesof the interestrate of BIC basedon the Wang's methodand proposedmehod respectively.
Fig. 4. The predictionand true valueso{ interestrate of BIC basedon the proposed
methodusing:(a) minimum inferenceengine;(b) product inferenceengine
trai
nOlYr
Iffi
[ lll Yo
coE
Jwmal of Suantitat ve Methods, Yo.S No.Z,December 2009, 7843
83
5. Conclusions
qaining dara. The
Jn this papcr, we t-ravgpreseft€d a qew method f-or higtring csrnplet€ fu_zz-yrules _f,nq!r.l
msthod usestlre firing strenglh of rule to construct complete fuzzy rules. The resulted fuzzryrules can cover all
values in the input spaces.The set of the fuzzy rules generatedby the proposedmethod contains all fuzzy nrtes
designed by the Wang's method In other wod the proposed method is generalization of the Wang's method.
We apply the pfqposedmethod b f-orecastthe Indonesianinflalion rate and intgr€st ra& of BIC. The result is tha!
fuecasting Indonesian inflation rde and interest rate of BIC using the proposed method has a htgher apcuracy
dun that using the Wang's method. To increasethe prediction accuracy, we san define more fuzzJ sets in the
input and output spacesbut defining more fuzzryset$en imply complexity of computations. So determining the
qpttmal t!um&.r of fivzy rules is irnpqlt4rlt !o get efficient c_ottlput4tiols.In the qsxt works, we will @ig1 the
optimal number of fuzzy rules.
6. Acknowledgoment
The authors would like to thank DP2M DIKTI and LPFM Gadjah Mada University for the financial support of
this ree€arch.
References
[1] Abadi, A-M., Suhnar, Widodo and SaleL S. (2006). Fuuy model for ctimaling inflation r&. Procceeding
o!Internatiorul Confererce on Matlematics arrd Natural kiences- Institut Teknologi Bandrng, Indonesia
Abadi,
A.M., Subanar,Widodo and Saleh, S. (2007). Fuecasting int€rest rate of Bank Indonesia certificate
[2]
based on univariate fuzzy time serics,,Internatiotul Conferetre on Matlematics atd lts applicatiotu
SU MS, Gadjah Mada Univerplty, "ltrdsnesia
[3] Abadi, A.M., Subanar,Widodo and Saleh, S. (2008s). Constnrcting complete firzy rules of fuzzy modet
using singular value decomposi6on.Preeeding of Internaional Cogferenceon Mathemdics, Staistics and
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Ir
INTRUCTIONS FOR THE AUTHORS
Journal of Quantitative Methods acceptsthe manuscriptsin English. Three copiesof the
manuscriptshouldbesentto:
Dr.Jamal I.Daoud
FacultyofEngineering
of Science,
DeparEnent
lslamicUniversityMalaysia
International
Email:Jamal58@iiu.edu.my
Themanuscriptthatcontainsoriginalresearchhasapriority to bepublished.In additionto, surveyor
review about the new developmenton certain topics also be consideredto be published. All
manuscriptswill berefereedbeforebeingacceptedto bepublished.Themanuscriptsshouldbetlped
inMS wordformA4paper,singlespacing,left andrightmargin2.5cm,topandbottommarginalso
3
cm. Maximumpagesfor eachmanuscriptare20. It is alsoshouldcontainthetitle ( font I l, Time
New RomanCapitalletter andbold) which is simpleandrepresentthetopic, thenameof author(s),
the nameof the institution, abstract,the main manuscriptand references.The abstractshouldbe
typedwith font 10,TimeNew Roman, andthe sectionof the manuscriptshouldbe typedwith font
I 0, TimeNew Romanandbold. Themainmanuscriptpart shouldbe typedwith font 10,TimeNew
Roman.
STEP.STRESSTEST USING THE NON.IIOMOGENEOUS POISSONPROCESS
FOR REPAIRABLE SYSTEMSAND ITS APPLICATION IN AGRICULTURE
Xiao Yang' and Imad Khamis'
SouthernMissouriStateUniversity
'Departmentof Mathematics,
OneUniversityPlaza.CapeGirardeau,
MO-63701
testsis focusedon nonrepairablesystems
Abstract. Most of theavailableliteratureon step-stress.
with weibull or exponentialdistribution.In thispaperanaltemativeprospectiveis presentedfor stepstresstestson repairablesystems,in which eachexperimentalunit canbe repairedandreplacedinto
testsaftera failure. Thebasiclife distributioninvolvedis a non-homogeneous
PoissonProcess.The
PoissonProcess(CENHPP)Model will be proposed.
cumulativeExposureNon-Homogeneous
Inferentialprocedureandoptimumdesignwill be discussed
for simplestep-stress
accelerated
test
plans with fixed shapeparameter.And severalpotentialapplicationsof CENHPPmodels in
agriculturewill berepresented.
test;Non-homogeneous
PoissonProcess;repairable
Keywords: Accreditedlife test; step-stress
systems;
maximumlikelihood;optimumdesigrt
1. Introduction
Write your paperhere.Write your paperhere.Write your paperhere.Write your paperhere.Write
yourpaperhere.Write yourpaperhere.Writeyourpaperhere.
References
acceleratedlife-test
data:A new Approach.IEEE
[1] Tang,L.C. (1990).Analysisof step-stress
TrawactionsonReliability,45 (l), 60-74
step-stressacceleratedlifetestsfor
[2] Chung,S.W.andBai,D.S.(1977).OptimalDesignofSimple
logrrormal lifetime distributions, International Journal of Reliability, Quality, and Safett
Engineering,s,3 I 5-336