FUZZY MODEL FOR FORECASTING INTEREST RATE OF BANK INDONESIA CERTIFICATE

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1 he 3 rd Internatonal Conference on Quanttatve ethods ISBN Used n Econoc and Busness. June 6-8, 00 FUZZY ODEL FOR FORECASING INERES RAE OF BANK INDONESIA CERIFICAE Agus aan Abad, Subanar, Wdodo 3, Sasubar Saleh 4 Departent of atheatcs Educaton, Faculty of atheatcs and Natural Scences, Yogyakarta State Unversty, Indonesa Karangalang Yogyakarta 558,3 Departent of atheatcs, Faculty of atheatcs and Natural Scences, Gadah ada Unversty, Indonesa Sekp Utara, Bulaksuur Yogyakarta Departent of Econocs, Faculty of Econocs and Busness, Gadah ada Unversty, Indonesa Jl. Huanora, Bulaksuur Yogyakarta 558 E-al: aanabad@yal.co, subanar@yahoo.co, 3 wdodo_ath@yahoo.co, 4 huas@paue.ug.ac.d Abstract. In fuzzy odellng, Wang s ethod s a sple ethod that can be used to overcoe the conflctng rule by deternng each rule degree. he weakness of fuzzy odel based on the ethod s that the fuzzy relatons ay not be coplete so the fuzzy relatons can not cover all values n the doan. Generalzaton of the Wang s ethod has been developed to construct copletely fuzzy relatons. But ths ethod causes coplexly coputatons. Furtherore, predcton accuracy depends not only on fuzzy relatons but also on nput varables. hs paper presents a ethod to select nput varables and reduce fuzzy relatons to prove accuracy of predcton. hen, ths ethod s appled to forecast nterest rate of Bank Indonesa Certfcate (BIC). he predcton of nterest rate of BIC usng the proposed ethod has a hgher accuracy than that usng generalzed Wang s ethod. Keywords: fuzzy relaton, sngular value decoposton, QR factorzaton, nterest rate of BIC.. Introducton Fuzzy te seres s a dynac process wth lngustc values as ts observatons. Song and Chsso developed fuzzy te seres by fuzzy relatonal equaton usng adan s ethod (Song and Chsso, 993a). In ths odelng, deternng the fuzzy relatons needs large coputaton. hen, Song and Chsso (993b, 994) constructed frst order fuzzy te seres for te nvarant and te varant case. hs odel needs coplexty coputaton for fuzzy relatonal equaton. Furtherore, to overcoe the weakness of the odel, Chen desgned fuzzy te seres odel by clusterng of fuzzy relatons (Chen, 996). Hwang, et.al (998) constructed fuzzy te seres odel to forecast the enrollent n Alabaa Unversty. Fuzzy te seres odel based on heurstc odel gave ore accuracy than ts odel desgned by soe prevous researchers (Huarng, 00). hen, forecastng for enrollent n Alabaa Unversty based on hgh order fuzzy te seres resulted ore accuracy predcton (Chen, 00). Frst order fuzzy te seres odel was also developed by Sah and Degtarev (004) and Chen and Hsu (004). Abad, et al (007) constructed fuzzy te seres odel usng table lookup schee (Wang s ethod) to forecast nterest rate of Bank Indonesa certfcate (BIC) and the result gave hgh accuracy. hen, forecastng nflaton rate usng sngular value decoposton ethod had a hgher accuracy than that usng Wang s ethod (Abad, et al, 008a, 008b). he weakness of the constructng fuzzy relatons based on the Wang s ethod s that the fuzzy relatons ay not be coplete so the fuzzy relatons can not cover all values n the doan. o overcoe ths weakness, Abad, et al (008c) desgned generalzed Wang s ethod. Furtherore, Abad, et al (009) constructed coplete fuzzy relatons of fuzzy te seres odel based on tranng data. oo any fuzzy relatons result coplex coputatons and too few fuzzy relatons cause less powerful of fuzzy te seres odel n predcton accuracy. hen, predcton accuracy depends not only on fuzzy relatons but also on nput varables.

2 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 In ths paper, we wll desgn optal nput varables and fuzzy relatons of fuzzy te seres odel usng sngular value decoposton ethod to prove the predcton accuracy. hen, ts result s used to forecast nterest rate of BIC. he rest of ths paper s organzed as follows. In secton, we brefly revew the basc defntons of fuzzy te seres. In secton 3, we present a procedure to select nput varables. In secton 4, we present a ethod to reduce fuzzy relatons to prove predcton accuracy. In secton 5, we apply the proposed ethod to forecastng nterest rate of BIC. We also copare the proposed ethod wth the generalzed Wang s ethod n the forecastng nterest rate of BIC. Fnally, soe conclusons are dscussed n secton 6.. Fuzzy e Seres In ths secton, we ntroduce the followng defntons and propertes of fuzzy te seres referred fro Song and Chsso (993a). Defnton. Let Yt () R, t =..., 0,,,...,, be the unverse of dscourse on whch fuzzy sets f () t ( =,, 3,...) are defned and Ft () s the collecton of f () t, =,, 3,...,then Ft () s called fuzzy te seres on Yt, () t =..., 0,,, 3,... In the Defnton, Ft () can be consdered as a lngustc varable and f () t as the possble lngustc values of Ft (). he value of Ft () can be dfferent dependng on te t so Ft () s functon of te t. he followng procedure gves how to construct fuzzy te seres odel based on fuzzy relatonal equaton. Defnton. Let I and J be ndces sets for Ft ( ) and Ft () respectvely. If for any f() t Ft (), J, there exsts f( t ) Ft ( ), I such that there exsts a fuzzy relaton R( tt, ) and f( t) = f( t ) R( tt, ), Rtt (, ) = R( tt, ) where s unon operator, then R( t, t ) s called fuzzy relaton between Ft () and, Ft ( ). hs fuzzy relaton can be wrtten as Ft () = Ft ( ) Rtt (, ). () where s ax-n coposton. In the equaton (), we ust copute all values of fuzzy relatons R( tt, ) to deterne value of Ft (). Based on above defntons, concept for frst order and -order of fuzzy te seres can be defned. Defnton 3. If Ft () s caused by Ft ( ) only or by Ft ( ) or Ft ( ) or or Ft ( ), then the fuzzy relatonal equaton Ft ( ) = Ft ( ) R( tt, ) or Ft ( ) = ( Ft ( ) Ft ( )... Ft ( )) R0 ( tt, ) () s called frst order odel of Ft (). Defnton 4. If Ft () s caused by Ft ( ), Ft ( ),... and Ft ( ) sultaneously, then the fuzzy relatonal equaton Ft ( ) = ( Ft ( ) Ft ( )... Ft ( )) Ra ( tt, ) (3) s called -order odel of Ft (). Fro equatons () and (3), the fuzzy relatons Rtt (, ), R(, tt ), R(, tt ) are portant factors to a o desgn fuzzy te seres odel. Furtherore for the frst order odel of Ft (), for any f() t Ft (), J, there exsts f( t ) Ft ( ), I such that there exsts fuzzy relatons R( tt, ) and f( t) = f( t ) R( tt, ). hs s equvalent to f f ( t ), then f () t, and then we have the fuzzy relaton R( tt, ) = f ( t ) f () t. Because of Rtt (, ) = R( tt, ), then, Rtt (, ) = aks {n( f ( t), f ( t ))}., (4) For the relaton R(, tt ) of the frst order odel, we get o R(, tt ) = aks{ aks{n( f ( t k), f ( t))} }. (5) Based on -order odel of Ft (), we have R(, tt ) = aks{ a o p,,,..., p k k, n ( k f ( t ) f ( t )... f ( t ) f ( t) )} (6) Fro equatons (4), (5) and (6), we can copute the fuzzy relatons usng ax-n coposton. Defnton 5. If for t t, Rt (, t ) = Rt (, t ) or Ra( t, t ) = Ra( t, t ) or R ( o t, ) t = R ( o t, ) t, then Ft () s called te-nvarant fuzzy te seres. Otherwse t s called te-varant fuzzy te seres.

3 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 3 e-nvarant fuzzy te seres odels are ndependent of te t. hose ply that n applcatons, the te-nvarant fuzzy te seres odels are spler than the te-varant fuzzy te seres odels. herefore t s necessary to derve propertes of te-nvarant fuzzy te seres odels. heore. If Ft () s fuzzy te seres and for any t, Ft () has only fnte eleents f () t, =,, 3,..., n, and Ft () = Ft ( ), then Ft () s a te-nvarant fuzzy te seres. heore. If Ft () s a te-nvarant fuzzy te seres, then Rtt (, )... f ( t ) f ( t) f ( t ) f ( t ) =... 0 f ( t ) f ( t + )... where s a postve nteger and each par of fuzzy sets s dfferent. Based on the heore, we should not calculate fuzzy relatons for all possble pars. We need only to use one possble par of the eleent of Ft () and Ft ( ) wth all possble t s. hs ples that to construct tenvarant fuzzy te seres odel, we need only one observaton for every t and we set fuzzy relatons for every par of observatons n the dfferent of te t. hen unon of the fuzzy relatons results a fuzzy relaton for the odel. heore s very useful because we soete have only one observaton n every te t. Let F() t be fuzzy te seres on Yt. () If F() t s caused by ( F( t ), F ( t )), then ( ) F ( t ) (, F ( t )),..., ( F( t n), F ( t n)), ( F( t n), F ( t n)),..., ( F t, F ( t )), ( F( t ), F ( t )) F() t s the fuzzy relaton and t s called two-factor n-order fuzzy te seres forecastng odel, where F (), t F () t are called the an factor and the secondary factor fuzzy te seres respectvely. If a fuzzy relaton s presented as ( F( t n), F ( t n),..., F ( t n)),..., ( F ( t ), F ( t ),..., F ( t )), ( F( t ), F ( t ),..., F ( t )) (7) then the fuzzy relaton s called -factor n-order fuzzy te seres forecastng odel, where F () t are called the an factor fuzzy te seres and F ( t ),..., F ( t ) are called the secondary factor fuzzy te seres. A ( t ),..., A ( t ) be N fuzzy sets wth contnuous ebershp functon that are noral and Let, k N, k coplete n fuzzy te seres F ( t ), =,, 3,, n, k =,,,, then the fuzzy rule: k R :,, IF ( x ( t n) s A ( t n) and...and x ( t n) s A ( t n)) and and ( x ( t ) s A ( t ) and...and x ( t ) s A ( t )), HEN,, x ()s t A () t (8), s equvalent to the fuzzy relaton (7) and vce versa. So (8) can be vewed as fuzzy relaton n U V where n U = U... U R, n V R wth µ A( x( t n),..., x( t ),..., x( t n),..., x( t )) = µ ( x ( t n))... µ ( x ( t ))... µ ( x ( t n)... µ ( t ), A A A A,,,, where A = (... ( )... ( )... ( ),,,, We refer to Abad, et al (009) to desgn -factor one-order te nvarant fuzzy te seres odel usng generalzed Wang s ethod. But ths ethod can be generalzed to -factor n-order fuzzy te seres odel. Suppose we are gven the followng N tranng data: ( x ( t ), x ( t ),..., x ( t ); x ( t)), p p p p p =,,3,..., N. Based on a ethod developed Abad, et al (009), we have coplete fuzzy relatons desgned fro tranng data: R l l l l l : ( A ( t ), A ( t ),..., A ( t )) A * ( t), l =,, 3,,. (9),,,, If we are gven nput fuzzy set A ( t ), then the ebershp functon of the forecastng output A () t s. µ ( ( )) A x t = () t ax (sup( µ ( ( )) ( ( )) ( ( )))) A xt µ x t µ A ( ) l x t. (0) f t f f A l= xu f =,, For exaple, f gven the nput fuzzy set A ( t ) wth Gaussan ebershp functon * ( x ( t ) x ( t )) µ ( ( )) exp( ) A xt ( t = ), then the forecastng real output wth center average defuzzfer s = a * ( x ( t ) x ( t )) y exp( ) = = a + σ, x ( t) = f( x ( t ),..., x ( t )) = () * ( x ( t ) x ( t )) exp( ) = = a + σ, where y s center of the fuzzy set A,() t.

4 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, Selecton of Input Varables 3. Senstvty of nput varables Gven fuzzy relatons where the l th fuzzy relaton s expressed by: n If x s A and x s A and... and x n n s A, then s and the output of fuzzy odel s defned by wth yt () = wy() t r r= r r= r r r n y r s output of r th fuzzy relaton, wr = A A... A, and r r ( x x ) A ( x) = exp( ) σ Saez, D, and Cprano, A (00) defned the senstvty of nput varable If σr = σ for every r, then ( x ) ( x ) ( ) ( ) ξ ( x) = wr r= n r ( x x ) wr = exp( ) σ wth = Senstvty ξ ( x) we copute r r w r x by y B yt () ξ ( x) =. x r r yw r r wr w r wy r r r= σ r= r= σ r= depends on nput varable x and coputng of the senstvty based on tranng data. hus, I = µ + σ for each varable where µ and σ are ean and standard devaton of senstvty of varable x respectvely. hen, nput varable wth the sallest value I s dscarded. Based on ths procedure, to choose the portant nput varables, we ust take soe varables havng the bggest values I. 3. Senstvty atrx In ths secton, we propose a ethod to select portant nput varables usng senstvty atrx. he portant nput varables can be detected by sngular value decoposton and QR factorzaton of senstvty atrx. Sngular value decoposton and QR factorzaton ethods are used to know strongly ndependence coluns referrng to Golub (976). Suppose, gven N tranng data and n nput varables, then selecton of nput varables can be done by the followng steps: Step. Copute senstvty of nput varables x as ξ ( x). Step. Construct senstvty atrx N x n as ξ() ξ() ξ () n ξ() ξ() ξn () s = ξ( N) ξ( N) ξn ( N) Step 3. Copute sngular value decoposton of s as = USV, where U and V are N x N and n x n orthogonal atrces respectvely, S s N x n atrx whose entres s = 0,, s = σ =,,..., k wth σ σ... σ k 0, k n( Nn, ). Step 4. Deterne the bggest r sngular values that wll be taken as r wth r rank( s ). s

5 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 5 V V Step 5. Partton V as V =, where V s r x r atrx, V s (n-r) x r atrx. V V Step 6. Construct V ( V V ) =. Step 7. Apply QR-factorzaton to V and fnd n x n perutaton atrx E such that V E = QR where Q s r x r orthogonal atrx, R = [R R ], and R s r x r upper trangular atrx. Step 8. Assgn the poston of entres one s n the frst r coluns of atrx E that ndcate the poston of the r ost portant nput varables. Step 9. Construct fuzzy te seres forecastng odel (0) or () usng the r ost portant nput varables. Step 0. If the odel s optal, then stop. If It s not yet optal, then go to Step Reducton of Fuzzy Relatons If the nuber of tranng data s large, then the nuber of fuzzy relatons ay be large too. So ncreasng the nuber of fuzzy relatons wll add the coplexty of coputatons. o overcoe that, frst we construct coplete fuzzy relatons usng generalzed Wang s ethod referred fro Abad, et al (009) and then we wll apply QR factorzaton ethod to reduce the fuzzy relatons usng the followng steps. Step. Set up the frng strength of the fuzzy relaton (9) for each tranng datu (x;y) = ( x ( t ), x ( t ),..., x ( t ); x ( t)) as follows L l (x;y) = µ ( x ( t )) µ ( x ( t)) A, ( ) l f t f f A f =, µ ( x ( t )) µ ( x ( t)) k= f = A, ( ) k f t f f A, L () L () L () () () () Step. Construct N x atrx L = L L L. L ( N) L ( N) L ( N) Step 3. Copute sngular value decoposton of L as L = USV, where U and V are N x N and x orthogonal atrces respectvely, S s N x atrx whose entres s = 0,, s = σ =,,..., r wth σ σ... σ 0 r, r n( N, ). Step 4. Deterne the bggest r sngular values that wll be taken as r wth r rank( L). V V Step 5. Partton V as V =, where V V V s r x r atrx, V s (-r) x r atrx, and construct V = V V. ( ) Step 6. Apply QR-factorzaton to V and fnd x perutaton atrx E such that V E = QR where Q s r x r orthogonal atrx, R = [R R ], and R s r x r upper trangular atrx. Step 7. Assgn the poston of entres one s n the frst r coluns of atrx E that ndcate the poston of the r ost portant fuzzy relatons. Step 8. Construct fuzzy te seres forecastng odel (0) or () usng the r ost portant fuzzy relatons. 5. Applcaton of the Proposed ethod In ths secton, we apply the proposed ethod to forecast nterest rate of BIC. Frst, we apply senstvty atrx ethod to select nput varables. Second, we apply sngular value decoposton ethod to select the optal fuzzy relatons. We consder the ntal fuzzy odel wth eght nput varables (x(k-8), x(k-7),, x(k- )) fro data of nterest rate of BIC. he data are taken fro January 999 to February 003. he data fro January 999 to ay 00 are used to tranng and the data fro June 00 to February 003 are used to testng. We use [0, 40] as unverse of dscourse of eght nputs and one output and we defne seven fuzzy sets A, A,..., A wth Gaussan ebershp functon on each unverse of dscourse of nput and output. hen, we 7 copute senstvty of nput varables and senstvty atrx usng the procedure n secton 3. and secton 3.. Dstrbutons of senstvty of nput varables and sngular values of senstvty atrx are showed n Fgure. We choose the bggest two sngular values and three sngular values and then we apply QR factorzaton to get the sgnfcant nput varables. Based on selectng the bggest two sngular values and three sngular values, the selected nput varables are x(k-8), x(k-) and x(k-8), x(k-3), x(k-) respectvely. hen, fuzzy te seres odel constructed by two nput varables x(k-8) and x(k-) has better predcton accuracy than fuzzy te seres odel constructed by three nput varables x(k-8), x(k-3), x(k-). So we choose x(k-8) and x(k-) as nput varables to

6 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 6 predct value x(k). hen we apply the generalzed Wang s ethod to yeld forty nne fuzzy relatons showed n able. (a) (b) Fgure. (a) Dstrbuton of senstvty of nput varables, (b) Dstrbuton of sngular values of senstvty atrx We apply the sngular value decoposton ethod n secton 4 to get optal fuzzy relatons. he sngular values of frng strength atrx are shown n Fgure. After we take the r ost portant fuzzy relatons, we get ten fuzzy relatons that are the optal nuber of fuzzy relatons. he postons of the ten ost portant fuzzy relatons are known as,, 8, 9, 0, 5, 7, 9, 37, 44 (blue colored rules n able ). he resulted fuzzy relatons are used to desgn fuzzy te seres forecastng odel (0) or (). able. Fuzzy relaton groups for nterest rate of BIC usng generalzed Wang s ethod Rule ( x ( t 8), x ( t ) ) xt () Rule ( x ( t 8), x ( t ) ) xt () Rule ( x ( t 8), x ( t ) ) xt () (A, A) A 7 (A3, A3) A 33 (A5, A5) A (A, A) A 8 (A3, A4) A 34 (A5, A6) A 3 (A, A3) A 9 (A3, A5) A 35 (A5, A7) A 4 (A, A4) A3 0 (A3, A6) A 36 (A6, A) A 5 (A, A5) A3 (A3, A7) A 37 (A6, A) A 6 (A, A6) A3 (A4, A) A 38 (A6, A3) A 7 (A, A7) A3 3 (A4, A) A 39 (A6, A4) A 8 (A, A) A4 (A4, A3) A 40 (A6, A5) A 9 (A, A) A 5 (A4, A4) A 4 (A6, A6) A 0 (A, A3) A3 6 (A4, A5) A 4 (A6, A7) A (A, A4) A3 7 (A4, A6) A 43 (A7, A) A (A, A5) A3 8 (A4, A7) A 44 (A7, A) A 3 (A, A6) A3 9 (A5, A) A 45 (A7, A3) A 4 (A, A7) A3 30 (A5, A) A 46 (A7, A4) A 5 (A3, A) A 3 (A5, A3) A 47 (A7, A5) A 6 (A3, A) A 3 (A5, A4) A 48 (A7, A6) A 49 (A7, A7) A Fgure. Dstrbuton of sngular values of frng strength atrx

7 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 7 Based on the able, the average forecastng errors of nterest rate of BIC usng the proposed ethod and the generalzed Wang s ethod are.8787%, % respectvely. So we can conclude that forecastng nterest rate of BIC usng the proposed ethod results ore accuracy than that usng the generalzed Wang s ethod. able. Coparson of average forecastng errors of nterest rate of BIC fro the dfferent ethods ethod Nuber of fuzzy relatons SE of tranng data SE of testng data Average forecastng errors (%) Proposed ethod Generalzed Wang s ethod he coparson of predcton and true values of nterest rate of BIC usng the generalzed Wang s ethod and the proposed ethod s shown n Fgure 3. (a) (b) Fgure 3. Predcton and true values of nterest rate of BIC usng: (a) proposed ethod, (b) generalzed Wang s ethod 6. Conclusons In ths paper, we have presented a ethod to select nput varables and reduce fuzzy relatons of fuzzy te seres odel based on the tranng data. he ethod s used to get sgnfcant nput varables and optal nuber of fuzzy relatons. We appled the proposed ethod to forecast the nterest rate of BIC. he result s that forecastng nterest rate of BIC usng the proposed ethod has a hgher accuracy than that usng generalzed Wang s ethod. he precson of forecastng depends also on deternng nuber of fuzzy sets and paraeter of fuzzy sets. In the future works, we wll construct a procedure to deterne the optal fuzzy sets to prove predcton accuracy. References [] Abad, A.., Subanar, Wdodo, and Saleh, S.(007). Forecastng nterest rate of Bank Indonesa certfcate based on unvarate fuzzy te seres. Internatonal Conference on atheatcs and Its applcatons SEAS, Gadah ada Unversty, Yogyakarta. [] Abad, A.., Subanar, Wdodo, and Saleh, S. (008a). Constructng coplete fuzzy rules of fuzzy odel usng sngular value decoposton. Proceedng of Internatonal Conference on atheatcs, Statstcs and Applcatons (ICSA), Syah Kuala Unversty, Banda Aceh. [3] Abad, A.., Subanar, Wdodo, and Saleh, S. (008b). Desgnng fuzzy te seres odel and ts applcaton to forecastng nflaton rate. 7 h World Congress n Probablty and Statstcs, Natonal Unversty of Sngapore, Sngapore. [4] Abad, A.., Subanar, Wdodo, and Saleh, S. (008c). A new ethod for generatng fuzzy rule fro tranng data and ts applcaton n fnancal probles. he Proceedngs of he 3 rd Internatonal Conference on atheatcs and Statstcs (ICoS-3), Insttut Pertanan Bogor, Bogor. [5] Abad, A.., Subanar, Wdodo, and Saleh, S. (009). Desgnng fuzzy te seres odel usng generalzed Wang s ethod and ts applcaton to forecastng nterest rate of Bank Indonesa certfcate. Proceedngs of he Frst Internatonal Senar on Scence and echnology, Islac Unversty of Indonesa, Yogyakarta. [6] Chen, S... (996). Forecastng enrollents based on fuzzy te seres. Fuzzy Sets and Systes, 8, pp.3-39.

8 he 3 rd Internatonal conference on Quanttatve ethods used n Econocs and Busness, 00 8 [7] Chen, S... (00). Forecastng enrollents based on hgh-order fuzzy te seres. Cybernetcs and Systes Journal, 33, pp. -6. [8] Chen, S.., and Hsu, C.C. (004). A new ethod to forecastng enrollents usng fuzzy te seres. Internatonal Journal of Appled Scences and Engneerng, (3), pp [9] Golub, G.H., Klea, V. and Stewart, G.W. (976). Rank degeneracy and least squares probles, ech Rep R-456, Dept. Coput. Sc., Unv. aryland, Colellege Park. [0] Huarng, K. (00). Heurstc odels of fuzzy te seres for forecastng. Fuzzy Sets and Systes, 3, pp [] Hwang, J.R., Chen, S.., and Lee, C.H. (998). Handlng forecastng probles usng fuzzy te seres. Fuzzy Sets and Systes, 00, pp.7-8. [] Saez, D., and Cprano, A. (00). A new ethod for structure dentfcaton of fuzzy odels and ts applcaton to a cobned cycle power plant. Engneerng Intellgent Systes,, pp [3] Sah,., and Degtarev, K.Y. (004). Forecastng enrollents odel Based on frst-order fuzzy te seres. ransacton on Engneerng, Coputng and echnology VI, Enforatka VI, pp [4] Song, Q., and Chsso, B.S. (993a). Forecastng enrollents wth fuzzy te seres, Part I. Fuzzy Sets and Systes, 54, pp. -9. [5] Song, Q., and Chsso, B.S. (993b). Fuzzy te seres and ts odels. Fuzzy Sets and Systes, 54, pp [6] Song, Q., and Chsso, B.S. (994). Forecastng enrollents wth fuzzy te seres, Part II. Fuzzy Sets and Systes, 6, pp. -8. [7] Wang, L.X. (997). A Course n Fuzzy Systes and Control, Prentce-Hall, Inc., Upper Saddle Rver.