voLUME 5, I\UMBER 2DECEMBER 2OO9

JOURNAL OF

rssN 1693-5098

|I UA]IIITAIIUE M ETH || II S

JOT]RNAL DEVOTED TO TIIE MATHEMATICALAND SIATISTICAL APPLICATIONS IN VARIOUSFIELDS

QUANTITATIVB METHODS

QUANTITATIVE METHODS provide a forum for communication between statisticians andmathematicians with the users of applied of statistical and mathematical techniques among a widerange of areas such as operations research, computing, actuary, engineering, physics, biology,medicine, agriculture, environment, management, business, economics, politics, sociology, andeducation. The journal will emphasize on both the relevance of numerical techniques and quantitativeideas in applied areas and theoretical development, which clearly demonstrate significant, appliedpotential.

Editor in Chief:Dr. Noor Akma Ibrahim

,,,,?1?ffffii;"j#ffiffi '1ii#lJl'.1'ii:J$?,HrJifr '.*"Editorial Assistant:Dr. Jamal I DaoudDepartment of Sciences, Faculty of EngineeringInternational Islamic Universitv Malavsia

Editors:Dr.ZhenbngZeng Dr.FaizA.M. ElfakiShanghai Institute for Biological Sciences Departrnent of SciencesChinese Academy of Science Intemational Islamic University, Malaysia

Department of Mathematics Faculty of Planning and ManagementUniversity Malaysia Trengganu, Malaysia Al Balqa, Applied University, Al-Salt, Jordan

Chen-ju Lin, Ph.D Dr. Hon Wai LeongDept. of Industrial Engineering&Management Department of MathematicsYuanZe University, Taiwan National University of Singapore, Singapore

Dr.Raihan OthmanDepartment of ScienceInternational Islamic University, Malaysia

Dr. Ismail Bin Mohd

Dr.Pieter N.JuhoDepartment of Mathematics,Natal University, South Africa

Dr. Sannay MohamadDepartment of MathematicsUniversity Brunei Darussalam,Brunei Darussalam

Dr. Mat Rofa IsmailDepartment of MathematicsUniversity Putra Malaysia, Malaysia

Dr.Maher Taka

Dr.Muhammad Azam KhanUniversity of Science and Technology Bannu,NWFP-Pakistan

Dr.Zainodin Haji JubokCentre for Academic AdvancementUniversity Malaysia Sabah,Malaysia

Dr. Andreas Dress Dr.Mustofa UsmanMax Plank Institute or Math. in the Sciences Department of MathematicsInselstrasse 22-26, D04103 Leipzig, Germany University of Lampung, Indonesia

Vol. 5, No.2 December 2009 QUANTITATIVE METHODS ISSN 1693-5098

TABLE OF CONTENTS

Data Envelopment Analysis For Stocks Selection On Bursa Malaysia...Ong Poay Ling, Anton Abdulbasah Karnil, Mohd Nain Hj. Awang

On Group Method Of Data Handling (GIDID ForData Mining.......Iing Lukman, Noor Akma lbrahim, Indah Lia Puspita and Eka Sariningsih

Stem Biomass Estimation of Roystonea Regia Using Multiple RegressionNoraini Abdullah, ZainodinHj. Jubok, Amran Ahmed.

Continued Fraction and Diophantine Equation..........Yusuf Hartono

Risk Analysis of Traffic And Revenue Forecasts For Toll Road InvestmentIn Indonesia .................

I Rudy Hermawan K., Ade Sjafruddin, dan Weka Indra Dharmawan

Use Of Ranks For Testing Fixed Treatment Effects In Basic Latin Square Design............Sigit Nugroho

Analysis of Good corporate Governance structure Effects on Bond Rating.......Einde Evana, Agrianti KSA, and R. Weddie Andryanto

A New Method for Generating Fuzry Rules from Training Data and Its Application ToForecasting Inflation Rate and Interest Rate of Bank fndonesia Certificate.

Agus Maman Abadi, Subanar, Widodoand Samsubar Saleh

Mean-var Portfolio Optimization Under Capm with Lagged, Non Constant Volatiltyand rhe Long Memory Erfect ....i;k;;;ffi;;;;.ilffi;ilr;dt

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JOURNAL OF QUANTITATIVE METHODSVol. 5, No. 2, Dec€mber 2009

rssN 1693-5098decqthere

A IYEW METHOD FOR GEI\TERATING WZZ,V RTJLES FROM TRAINING DATAAND ITS APPLICA.TION TO FORECASTING II\TFLATION RATE AIYI} INTEREST

RATE OF BAhiK INilX}IYESIA CERTIFICATE

Agus Maman Abadir, subapaf, widodq?and slmsubar sarehr

'Departnent of Mathematics Fsculty of Mathemstics and Nahnal Sciences,Yogyakata State LJniversity, Indonesia

2Departrnentof ffiffi13:r:ffiffi ffiffi *t*rii"o'ulsciences.Gedjah Mads Univeaslty, Indonesia

3Department of ,.*"ff*rYffi tdl#ffi ":Hrffitr;*j;$ff ifrfi u*"^ity, rndonesia

JL Humaniora" Bulaksumur yoryakarh 552g1, Iodonesia

Email: rmamanabadi@.vmail.corn- 2subanar@vahoo.conr. 2widodo-math@vahoo,conr. shumas(opaue.uern.ac,id

(Received l0 April 2@9, accepted 20 Octobsr 2009, published Decernber 20@)

Abstract' Table tookup scheme is a simple method to construct fuzzy rules at fuzzy model. That can be used toovercome the conflicting ryle by determining each rule degree. The weakness of'furzy model based on tablelookup scheme is that the fuzzy rules may not be complete so the fuzzy rules can not cover all values in thedomain' In this paper a new method to generate fuzzy rules from traininj oata wiil k proposed. In this method,all complete fuzzy rules are identified by firing strength of each possibie fuzzy rule."rhen, the resulged furzyrules are.used to dosignfuzzy mo&l- Applicati,ons oittr" p.opor"o method to predict the Indonesian inflationrale and interest rate of Bank Indonesia certificate {BIC) wilf be discussed. Thb predictioas of the Indonesianinflation rate and interest rate of BIC using the proposed method have a higher accuracy than those using thetable lookup scheme.

Keywords: fi'uzy rule, fuzzy model, firing strength of rule, inflation rate, interest rate of BIC.

l. Introduetion

Designing fuz4 rule base is one ,of some steps in fuzzy modelin g. Fuzzy rule base is the heart of ruzzy model.Recently, fuzzy model was developed by some researchers. waig, L.x. (1997) created fuzzy model based ontable lookup scheme, gradient descent haining, recursive least squires and clustering methods. The weakness ofthe fuzzy model based on table tookup scheme is that the fuz4 rulebase may not b€ complete so the fuzzy rulecan not cover all values in the domain' wu, T-P, and Cben, -S.M.

(ts9S) &sigred mem'terstrip functions andfuzzy rules from training data using a -cut of fuzzy sets. In wu and Chen's .Ltto4 determining membershipfunctions and fvzzy rules. needs large cromputations. To reduce the complexity of computation, iu*, y., et at(1999) built a method to decrease fuzzy rules using singular value decomposition.

fen, J', et al (1998) developed fuzzy model by combining globat and local learning to improve theinterprelability. New form of fuzry inference systems that was h-igily interpretable based on-a ludlcious choiceof membership function was presented by Bikdash, M. (1999). C".."., LJ., et al (2005) constructed Takagi-Sugeno-Kang models having high interpretability using Taylor series approximation. Thin, pomares, H., et al(2002) identified structure of fuzzy systems based on

"o*pi*" rules thai can decide which inpui uariattes .-nust

be taken into account in the fuzzy system and how many membenhip functions are needed in every selectedinput variable- Abadi' A.M., et al (200sa) designed complete fuzzy rutes using singular value decomposition. Inthis method, the prediction enor of training data depended only on taking the n-umdr of singular naiues.

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) showed that forecasting the lndoneJian inflat*ion *t" by:fuzzy model resultsd more accuracy than that by regression mcthod. Then, Abii, A.M., et al (2007) constuctedfuzzy time series model using tablc lookup scheme to forecast the interest rate of Bank Indonesia certificate(BIC) and its result gave high accuraoy. Then, Abadi, A.M., et al (2008b) showed that forecasting Indonesianinflation rate based on fuzzy timc series data using combination or talte lookup scheme and singular value

accutt90t!!dIndooWang

The nfiErrule hintfi!rate N

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the follor

Jownal of Quan|itdive Metho&, Va5 No'2, December 2009, 7833 79

decompcition methods had a higher accufttcy than that using Wang's method- Based on the previous research'

there are irreresting topics in frzzy model especially in determining fuuy rules that give good prediction

accuracy. To oveircome the weakness of tabll bokup scheme (Wang's method), in tbis paper, we design

;*ui; n rw *rer tI {wq *oa"l *ing-ruing strength of rule Ta.thu! its rcsult is uspd to predict thg

Indonesian inflation rate andinterest rat€ ;f BIC. The pmposed method has higher prediction accuracy than

W*g', method in its applications to predicting the lndonesian inflation rate and interest rate of BIC'

The rest of this pper is organized as follorrs. In section 2, we briefly review the Wang's me{hod to coostruct

n '"y

*oa.f. In'xfoon 3, we pr65ent a new method to constmct complete fuzzy rules using.filng strenglh of

rule bssed ur training OaL fn section 4, we apply tlre proposed_'n-.9d to forecasting the inflation rate and

inter€st rate of BIC. We also compare the propoied mettroO wittr the Wang's method in the forecasting inflation

rate and intp.rest 1-3tp of BIC, FiqAlly, Some conclusions q1p dispussod in section 5'

2. Wang's method fordesigning fuzzy rules

In this sestior," we will introduce the wang's method to consEuct fuzzy rules referred from wang, L.X. (1997).

Suppose tlrat we are given the following l{ input<nrtput dulai (x,o,x2r,--,x,ri!o), p=loZ'3,"',il where

x,, efa,, F,lc R arrd % e1ur, FrlcR , i: 1, 2, "', n'Designing fuzzy model using Wang's method is given

by the folloudng stePs:Siep l, Oefins &zzJ sets to gorer the input and output domains'

Foreachspace [a, , g, l , i :1,2, . . . ,n,def ine N,f i tzrzysets l , i , i : l r2, . . . ,Niwhicharecomplete in fa, ,p, l '

Similarly, define N, fuzy ss{s Br ,i = l,2o .-.,Nv which are complek in lar, f ,7'

Stcp 2. Generate one rule from one input-output pair'

For each input-ou9ut pair(r,r,rrr,. ..,x,.;!r). determine the membership value of x,r, I : 1,2' "', n in fiz4

sets 4 , j = l,2, ..., Ntand membership value of yrin fuzzy sets Bi ,i = 1,2' "', N"' Then, for each input

variabler,r, i=l,2,...,n,determinethe fuzzysetinwhichr,, hasthe largcsmembershipvalue. Indherwor4

determine t such that tto(t,)2lt,(",), 'i =lo?,-.N,. Similarly, ddermine Bn such that

pd,(Jr)>lto(t),1=1,2,"-Nr'Finally,wcconstructafrvzyIF-THENrule:

IF4is 4andx, is. l ' and' . -andr, is l i ,THEN yisf t r ( l )

Stcp 3. Compute degree of each rule dqsigred in step 2'

frol step 2,'one rui-e is gener*ed by onJinput-ouput pair. If the number of input-output-data is large, then it is

possible itrat there ur" tti" conflicting rules. Two rules be.come conflicting rules if the rules have same IF parts

but different THEN parts. To resolie this pnoblem, we assigt a degree tro each rule designed in step 2' The

degree of rule is 'defined

as follows:- s-uppase the rules (l) is consuucted by the input-output

pair(x,o,xr",. .-x,rilr) 'then

its degrre is defined as

D(rute) = l] tt o G, o) a n (t,)

Stcp 4, Conshuct the fuzzy rule base.The rule base consists otitre following thre€ sets of rules: (1) The rules designed in Step 2 that do not conflict

with any other rules; tii rrt" -r"

froir a conflicting group that has the maximum d"se"; (3) Linguistic rules

from human exPerts.Step 5. Consnuct the fuzzy model using thefiuzy rule base'

We can use any nofi"ri t*ty inferelnce engine and defuzzifiet combined with the fuzzy rule base to desigrr

fuzzy model. ti ttt" nurnU"t of raining data I N and the number of all possible combinations of the fuzzy sets

defined for the input variables is l-t N, , then the number of fuzzy rules generated by Wang's method may be

much less than both ^/

and fl{ . Then, the fuzzy rule base generated by this method may not be complete so

that the fuzzy rules can not cover all values in the input spaces. To overcome this weakness, we will design the

fu2ry des covering all values in input sipaces'

3. New method for constructing fury rules

Given the following N training datlt (x,o,xrr,- '.x,0;!o), p=1,2,3,"',N where x,uela,'p,l cRand

y, eld,, 0]c n, i: 1,2,..., r. we will inhoduce a new method to design fuzzy des. The method is given by

the following stePs:

Jownal of euantitative Metho*, vo.s No.2, Deceuber 2009, zg{.3 g0Step 1. Dsfine fuzzy sets to cover the input and output space&For each space [4,, f ,1, i = 1,2, ..., r, defrne N, fiEq *ts A! , j = 1,2,..., /{ which are complete and normalinfa,,f ,1. Similarly,define t{, fuzzysets Br,j=1,2,...,N, whicharecompleteandnormal infa,,frl.Step 2. Determine al! po. ssible antecedents of fuqy rulg candidates.Based on the Step I' there ut" nlf, antecedents of fuz4 rule candidates. The antecedent has form

"4 is 4n and x. is 4 nd... and x" is {' " simplified by * 4 and A! nd ...nd 4 ". For example, if we havetwo inpul and wp deline two fuzzy scts A, A. for first input spoee 4nd f,,, C, for second inpql spaee" then allpossibleantecedcnts of fitzy rule candidates ue A,andC,;.1 atdCr 4*dC,; \wrdCr.Step 3. Determine consequence of each fuzzy rule candidate.For each antecedent 4 nd At and ... and A!, the consequence of fuzzy rule is determined by firing strength ofthe rule uo&,)u*Qr)...P+Q,,)ttn(l)based on the training data- Choosing the consequence is done asfollows: For any Eaining data(xrr,xr,,...,x"ri-)rr)and for any tuzzy setgr, choosegr such thatIt*(x'e)lt*(x,r)...F4(x,".)tr",(ln)2 uo(\,)tro(x,,)...!t+(x^")ltn(l),forsome (4o.,rr,,,...,x,,.;yo.).

If there are at least tu'oBr such that uo(xrn\,.pe(x,,.)tturo) > p4!,(q)...rt*(x",)rtu,(%), then chooseone of 8'' . From this step, we have the fuzzy rule:

iF -r, is A( and x, 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.Step 4. Construct fuzzy rule base.

The fu:zy rule base is constructed by the fl l', fuzzy rules designed by 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 anddefuzzifier- For example, if we use singleton fuzzifier, produc,t inference engine and ont", uu"nag" defuzzifier,then the fuzzy modcl has form:

M ,Lb,l lp.,(x,)

f (x,xr,. . . ,x,)= 'n " j='

-Lp,ttr@,)

rvhere M is the number of rules.

lrom Step 3, the sel of fuzzy rules constructed by this me&od contains fuzzy rules designed by the Wang'smethod.'Iherefore the proposed method is the generalization of the wang's method.

4. Applications of the proposed method

In this section' we apply the proposed method to formast the Indorresian inflation rate and inter€st rate of BIC.The proposed method is implemented using Matlab 6.5.1 .

4.1 Forecasting inflation rate

The data ofinflation rate arelaken from January 1999 to January 2003. The data fiom January 1999 to March2002 are used to training and_the data from Aprit 2002 to January 2003 are used to testing. ihe procedure toforecasting inflation rate based on the proposed method is given byihe following steps:(l)' Define the universe of discourse of each input. In this paper, we will predict the inflation rate of month tusing inflation rate data of months &-t and &-2 so we will builA fuzzy model using 2-input and l-output. Theuniverso ofdiscourse ofeach input is defined as [-2, 4].(2). Define fuzzy sets on universe of discourse of each input such that fuzzy sets can cover the input spaces. wedsfine thirteen fuzzy sets 4,4,...,1\, that are normal and complete on [-2, 4l of each input space with Gaussianmembership function.(3). Determine all possible antecedents of fuzz1 rule candidates. There are 169 antecedents of fuzzy rulecandidates and the consequ€nce ofeach antecedent is chosen using Step 3 ofthe proposed method. So we have169 fu24 rules in the form:

R,: "lf -tr _2 is 4 *d .r*-, is 1b, tUeN xo is AJ.

wherr(4).o(5). 0defrrzdefuz

Table tmethodconstr[ltudfgfuzzifiG

Table tin{bneo.infereoaon the IFigure I

gl Josnal of gnntitdive Metho&, Yo.S No.2, December2h|g, Z8A3 gl

wherel :1,2, , . . "169, i t , iz= 1,2,. , . ,13 and AJ e14,4,- . ,4r1 .(4). Construct fuzy rule base frun frrzzy rules designed in St€p 3.(5). Oesigrr fu2ry model combining the fuzy rule basc and cartain fizzifier, fiuzy inference engine anddefuzafier. In this paper, we use singleton fuzdfrer, minimum and produst infererce engines, center averagedefuzzifier.

Teble l. Comparison of mean $quare €frors of f-orpeasting inflation rateusing the Wang's method and proposed method

Methods Numberoffuzzyrules

MSE of training data MSE of testing dataMinimuminferenceensine

Pmductinferenceeneine

Minimuminferenceensine

Productinferenceensine

Wang'smethod

23 0.47t60 0.45780 0.35309 0.35286

Proposedmethod

169 0.43436 0.407E7 o26097 0.26734

Table I shows a comparison of the MSE of training and testing data based on the Wang's me{hod and proposedmethod. There are twenty three fuzzy rules generated by the Wang's method and there are 169 fuzy-mlesconstructed by the proposed rnethod, The predistion of inllation rat€ by the Wang's method with singl*onfuzzifier, product inference engine and center average defirzzifier has a higher accuracy than that with singletonfitnifier, minimum inference engine and c€nt€r average defiuzifier.

Table I illushates that the forecasting inflation raie by the proposed method with singleton fiszzifrer, minimuminference engine and center average dcfuzzifier has a higher accuracy than t$at with singleton fv.ifier, productinference engine ald oent€r average defuzzifier. The true values and prediction values ofthe inflation rate basedon the Wang's method and the proposed method using minimum and product inference engines are shown inFigure I and Figure 2 resp@ively,

IE

!rt

rd[

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h

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IN

f

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f5

x

bcrr3rf ,

. lc

aiTht

Ftg. l. The prediction and true values ofinflation rate based onthe Wang's method using: (a) minimum inference engine;(b) product inference engine

Fig. 2. The prediction and fue values of inflation rate based onthe proposed method using: (a) minimum inference engine;(b) product inference engine

lm.

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brc

Journal of Quantitative Methods, Vo.S No.2, December 2009, 7843 tz

4.2 Forecasting interest rate of Bank Indonesia certificate

The data of the intercst rate of BIC fiom January 1999 to December ?00! afe us€d to Eerning flld th9 data fromJanuary 2002 to February 2003 are used to lesting. The interest rate of BIC of P month rvill be predicted by dataof interest rate of BIC of (t-l)* and (t-2)h months so we will build fuzzy model using 2-input and l-outputwhich universe ofdiscourse ofeach input is [10,401. Seven fuzzy sets are de{ined in dvery input space withGaussian membership function. The olher st€ps to forecasting the interest rate of BIC Use thq same steps of thepredicting inflation rate.

Teblc 2. Comparison of mean squar€ errors of predicting interest rate of BICusing thg Wang'$ melhod and proposgd method

Method

Numberof fuzzyrules

MSE of trainins data MSE of testine dafa Average forecastingerrors oftesting data

(o/o\Minirnuminference

engine

Productinferenceengine

Minimuminferenceengine

Productinferenceengine Minimum

int'erenceensine

Productinferenceensine

Wang'smethod

12 0.98759 1.0687 0.46438 0.55075 3.8568 4.4393

hoposedmethod

49 0.9536r 0.91682 o.2E3Z:7 0.24098 2.q973 2.7813

Table 2 shows that there are twelve fuzzy rules generated by Wang's method and there are forty nine fuzzy nrlesconstructed by proposed method. The alerage forecasting erors of the prediction of the interest rate of BIC byWang's method and proposed method are 3.8568 Vo and2.78l3 o/o respectively.

Fig. 3. the prediction and true values of interest rete of BIC based on the Wang's methodusing: (a) minimum inference engine; (b) product inference engine

From Table 2, forecasting the interest rate of BIC based on the Wang's method using minimurn inference enginehas a good accuracy. Based on the proposed metho4 forecasting of the interest rate of BIC using productinference engine has a good accuracy. The MSE values and average forecasting errors based on the proposedmethod are smaller than those based on the Wang's method, Figure 3 and Figure 4 show the true values andprediction values of the interest rate of BIC based on the Wang's method and proposed mehod respectively.

5. Cr

In thbmetiodvalues id"sipcWe apfor€calthan ftinput rotrimdoptirnd

6. Ach

The aufthis re*

Refera

l l l Ahqt

t2l AhbeJSFJ

t3l Ahunr,1p

[4] Ahtolsiq

[5] Bft!u

[6] Hcn427

lzl Pqfitza

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Fig. 4. The prediction and true values o{ interest rate of BIC based on the proposedmethod using: (a) minimum inference engine; (b) product inference engine

Jwmal of Suantitat ve Methods, Yo.S No.Z, December 2009, 7843

5. Conclusions

Jn this papcr, we t-ravg preseft€d a qew method f-or higtring csrnplet€ fu_zz-y rules _f,nq!r.l qaining dara. Themsthod uses tlre firing strenglh of rule to construct complete fuzzy rules. The resulted fuzzry rules can cover allvalues in the input spaces. The set of the fuzzy rules generated by the proposed method contains all fuzzy nrtesdesigned by the Wang's method In other wod the proposed method is generalization of the Wang's method.We apply the pfqposed method b f-orecast the Indonesian inflalion 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 apcuracydun that using the Wang's method. To increase the prediction accuracy, we san define more fuzzJ sets in theinput and output spaces but defining more fuzzry set$ en imply complexity of computations. So determining theqpttmal t!um&.r of fivzy rules is irnpqlt4rlt !o get efficient c_ottlput4tiols. In the qsxt works, we will @ig1 theoptimal number of fuzzy rules.

6. Acknowledgoment

The authors would like to thank DP2M DIKTI and LPFM Gadjah Mada University for the financial support ofthis ree€arch.

References

[1] Abadi, A-M., Suhnar, Widodo and SaleL S. (2006). Fuuy model for ctimaling inflation r&. Procceedingo!Internatiorul Confererce on Matlematics arrd Natural kiences- Institut Teknologi Bandrng, Indonesia

[2] Abadi, A.M., Subanar, Widodo and Saleh, S. (2007). Fuecasting int€rest rate of Bank Indonesia certificatebased on univariate fuzzy time serics,, Internatiotul Conferetre on Matlematics atd lts applicatiotuSU MS, Gadjah Mada Univerplty, "ltrdsnesia

[3] Abadi, A.M., Subanar, Widodo and Saleh, S. (2008s). Constnrcting complete firzy rules of fuzzy modetusing singular value decomposi6on. Preeeding of Internaional Cogference on Mathemdics, Staistics andApplicaions (ICMSA). Syiah Kuala University, Indonesia

[4] Abadi, A.M., Subonar, Widodo and Sale[ S. (2008b). Designing fuzzy time series model and its applicaticnto forecasting inflation rztr. 7'' World Congrets in Prabobility ctrd Stuistics. National University ofSingapce, Singapore

[5] Bikdas]t, M. (1999). A highly interpretable furm of Sugeno inlbrrcnc€ systems, IEEE Trarxrctions onfuzryEsfenr, 1(6),68ffi96

[6] Henera, LJ., Pomareg H., Rojas, I., Valensuela, O. and Prieto, A. (2005). Fuzzy sets and systems,l53, {)3-427

[{ Pomares, H., Rojas, I., Gonzales, J. and Prieto, A. (2002). Strusture identification in complete rulebasedfirzzy systcms - IEEE Transqtions onfto4t SJ,sterrs, l0(3), 349-359

[8] Wang t)( (1997). A cource lnfuzzy systems andcontral. Prentice-Hall, Inc, Upper Saddle River[9] Wq T.P. and Chen, S.M. (1999]. A new method for construc.ting membership fr.rnctions and rules from

haining examples. IEEE Transactiotls on Slstems, Maa and Cyherrctics-Part B: Cybenretics,29(l), 2540[10] Yam, Y., Baranyi, P. and Yan& C.T. (1999). Reduction of fuzzy rule base via singular value decomposition.

IEEE Trourctions on fuzry Slstems, 7 (2\, 120.132[ll] Yen, J., Wang L. and Gillespie, C.W. (1998).Improvingthe interpretability of TSK fuzzy models byN

combining global learning and local learning. IEEE Transrctions onfuzzlt Systems, 6{4),530-537

83

Ir

INTRUCTIONS FOR THE AUTHORS

Journal of Quantitative Methods accepts the manuscripts in English. Three copies of themanuscript should be sent to:

Dr.Jamal I.DaoudDeparEnent of Science, Faculty ofEngineeringInternational lslamic University MalaysiaEmail: Jamal5 8@iiu.edu.my

The manuscript that contains original research has a priority to be published. In addition to, survey orreview about the new development on certain topics also be considered to be published. Allmanuscripts will be refereed before being accepted to be published. The manuscripts should be tlpedinMS wordformA4paper, single spacing, left andrightmargin2.5 cm, top andbottommarginalso 3cm. Maximum pages for each manuscript are 20. It is also should contain the title ( font I l, TimeNew Roman Capital letter and bold) which is simple and represent the topic, the name of author(s),the name of the institution, abstract, the main manuscript and references. The abstract should betyped with font 10, Time New Roman, and the section of the manuscript should be typed with fontI 0, Time New Roman and bold. The main manuscript part should be typed with font 10, Time NewRoman.

STEP.STRESS TEST USING THE NON.IIOMOGENEOUS POISSON PROCESSFOR REPAIRABLE SYSTEMS AND ITS APPLICATION IN AGRICULTURE

Xiao Yang' and Imad Khamis''Department of Mathematics, Southern Missouri State University

One University Plaza. Cape Girardeau, MO-63701

Abstract. Most of the available literature on step-stress. tests is focused on non repairable systemswith weibull or exponential distribution. In this paper an altemative prospective is presented for step-stress tests on repairable systems, in which each experimental unit can be repaired and replaced intotests after a failure. The basic life distribution involved is a non-homogeneous Poisson Process. Thecumulative Exposure Non-Homogeneous Poisson Process (CENHPP) Model will be proposed.Inferential procedure and optimum design will be discussed for simple step-stress accelerated testplans with fixed shape parameter. And several potential applications of CENHPP models inagriculture will be represented.

Keywords: Accredited life test; step-stress test; Non-homogeneous Poisson Process; repairablesystems; maximum likelihood; optimum desigrt

1. IntroductionWrite your paper here. Write your paper here. Write your paper here. Write your paper here. Writeyour paper here. Write your paper here. Write your paper here.

References[1] Tang, L.C. (1990).Analysis of step-stress acceleratedlife-test data: A new Approach. IEEE

Traw actions on Reliability, 45 (l), 60-7 4[2] Chung, S.W. andBai,D.S. (1977). OptimalDesignofSimple step-stressacceleratedlifetestsfor

logrrormal lifetime distributions, International Journal of Reliability, Quality, and SafettEngineering,s, 3 I 5-336