Normal Type-2 Fuzzy Rational B-spline Curve

Transcription

1 ormal Type-2 Fuzzy Raonal B-splne Curve Rozam Zakara Deparmen of Mahemacs, Faculy of Scence and Technology, Unvers Malaysa Terengganu, Malaysa. Abd. Faah Wahab Deparmen of Mahemacs, Faculy of Scence and Technology, Unvers Malaysa Terengganu, Malaysa. R. U. Gobhaasan Deparmen of Mahemacs, Faculy of Scence and Technology, Unvers Malaysa Terengganu, Malaysa. Absrac In hs paper, we proposed a new form of ype-2 fuzzy daa pons(t2fds) ha s normal ype-2 daa pons(t2fds). These brand-new forms of daa were defned by usng he defnon of normal ype-2 rangular fuzzy number(t2tf). Then, we appled fuzzfcaon(alpha-cu) and ype-reducon processes owards T2FDs afer hey had been redefned based on he suaon of T2FDs. Furhermore, we redefne he defuzzfcaon defnon along wh he new defnons of fuzzfcaon process and ype-reducon mehod o oban crsp ype-2 fuzzy soluon daa pons. For all hese processes from he defnng he T2FDs o defuzzfcaon of T2FDs, we demonsrae hrough curve represenaon by usng he raonal B-splne curve funcon as he example form modelng hese T2FDs. Keywords: Type-2 fuzzy se, ype-2 fuzzy daa pons, alpha-cu, ype-reducon, defuzzfcaon. 1 Inroducon In order o consruc curves and surfaces, he collecons of daa are needed o

2 make he curves and surfaces can be modeled hrough her exsng funcons [5,8,10,11,14,15,21,22,24,26,42]. These consruced curves and surfaces become he represenave of he daa collecon o undersand of he daa condon and also make a concluson of he modelng daa. However, f he daa become unceran, subsequenly here exs a problem n modelng he uncerany daa hrough curves and surface. Therefore, by he defnon of ype-1 fuzzy se heory, hese knds of daa can be defned, whch also usng ype-1 fuzzy number and ype-1 fuzzy relaon and aferwards can be modeled hrough curves and surface [1,4,33,35,34,38,39,37,36,41,40]. However, n dealng he problem n defnng he complex uncerany daa, he exsng mehod of ype-1 fuzzy se heory(t1fst) s unable o apply n order o defne hem. Therefore, Zadeh comes wh he new heory whch known as ype-2 fuzzy se heory(t2fst) [44]. By usng hs heory and he concep of ype-2 fuzzy number(t2f), we can defne he complex uncerany daa whch become ype-2 fuzzy daa. For real daa problem, we also used T2FST, T2F and ype-2 fuzzy relaon(t2fr) o defne hem, whch become ype-2 fuzzy daa pons(t2fd). Due o he new defnon of defnng complex uncerany daa hrough T2FST, we wll dscuss abou he fuzzfcaon process(alpha-cu operaon) whch s been appled frs before we come o he nex mehod n geng he brand-new ype-2 fuzzy nerval pons from he old ype-2 fuzzy nerval pons of T2FDs. Then, we used he exsng ype-1 defuzzfcaon mehod o defuzzfy he new T2FDs for obanng crsp ype-2 fuzzy soluon daa pons. Ths paper organzed as follows: Secon 2 dscusses of some bascs defnons of T2FST. Then, n Secon 3, we dscuss abou he T2FDs defnon along wh he alpha-cu operaon process, ype-reducon and also defuzzfcaon mehods. For Secon 4, we wll dscuss abou he developmen of T2FDs based on he defnon of T2TF and he processes of geng crsp ype-2 fuzzy soluon daa pons before we demonsrae for all processes ha had been menoned before by usng he raonal B-splne curve funcon as he hypohecal example as n Secon 5. Then, hs proposed mehod can be summarzed n Secon 6. 2 Type-2 Fuzzy Se Theory T2FST was nroduced by Zadeh n 1975 [44] o solve he problem n defnng he complex uncerany whch he problem s unable o defne he exsng T1FST. Here, we wll defne some basc heores of T2FST, whch gven as follows. Defnon 2.1. A ype-2 fuzzy se(t2fs), denoed A, s characerzed by a ype-2 membershp funcon µ ( xu, ), where x X and u U [0,1] A x ha s, A= ( xu, ), µ ( xu, ) x X, u U [0,1] {( ) A } x

3 n whch, 0 µ ( xu, ) 1 [28]. A Defnon 2.2. A T2F s broadly defned as a T2FS ha has a numercal doman. An nerval T2FS s defned usng he followng four consrans, where A [ a, b ],[ c, d ] 0,1, a, b, c, d (Fg. 2.1) [25,45]: = { }, [ ] 1. a b c d 2. [ a, d ] and [ b, c ] generae a funcon ha s convex and [ a, d ] generae a funcon s normal , 2 [0,1]: ( 2 > 1) ( a, c a, c, b, d b, d, for c b. ) 4. If he maxmum of he membershp funcon generaed by [ b, c ] level, ha s, [ m m b, c ], hen m m m = 1 = 1 b, c a, d. s he a b c d 1 x 0 0 a b c d Fgure 2.1. Defnon of an nerval T2F. x Defnon 2.3. Le X, Y, Ux, Vy R and A= ( ( xu, ), µ ( xu, ) ) x X, u U [0,1] A x B= ( y, v ), µ ( yv, ) y Y, v V [0,1] { } {( ) B } y are wo T2FSs. Then, R= ((( xu) ( y v) ) µ µ (, ), µ ( yv, ) ( x X u U ), [0,1] R ( xu A B )) T2FR on A and and {,,,,, x ( y Y u V y ) } B (( xu, ),( yv, )) ( x X, u Ux) ( y Y, v Vy). 3 Type-2 Fuzzy Daa ons f ( ) s a ( xu,, ( y, v )) ( xu, ) µ µ µ µ, R A B A

4 Afer he T2FST, T2F and T2FR had been defned, hen we used hem o defne he complex uncerany daa whch wll become T2FDs s gven as follows. Defnon 3.1. Le = { x x ype-2 fuzzy pon} and = { daa pon} whch s se of ype-2 fuzzy daa pon wh X, where X s a unversal se and µ ( ) : [0,1] s he membershp funcon whch defned as µ ( ) = 1 =, µ ( ). Therefore, and formulaed by {( ) } 0 f X µ ( ) = c (0,1) f X (3.1) 1 f X wh µ ( ) µ ( = ), µ ( ), µ ( ) whch µ ( ) and µ ( ) are lef and rgh fooprn of membershp values wh µ ( ) = µ ( ), µ ( ), µ ( ) where, µ ( ), µ ( ) and µ ( ) are lef-lef, lef, rgh-lef membershp grade values and µ ( ), µ ( ) and µ ( ) are rgh-rgh, rgh, lef-rgh membershp grade values, whch can be wren as = : = 0,1, 2,..., n (3.2) { } for every, =,, wh =,, and where, are lef-lef, lef and rgh-lef T2FDs and =,, where, and are lef-rgh, rgh and rgh-rgh T2FDs respecvely. Ths can be llusraed as n Fg The llusraon of T2FD was shown n Fg. 3.1 whch T1FD becomes he prmary membershp funcon bounded by upper bound,,, and lower bound,,, respecvely. The process of defnng T2FD can be shown hrough Fg. 3.2.

5 u x Fgure 3.1. T2FD around 5. µ A ( x) µ ( x A ) µ A( x Ordnary daa pon x ) µ ( x) A Type-2 fuzzy daa pon x Crsp daa pon x Type-1 fuzzy daa pon Fgure 3.2. rocess of defnng T2FD. x Fg. 3.2 shows ha he process of defnng T2FD from he ordnary pon. Ths T2FD formed based on he defnon of T2F and T2FR. 4 ormal Type-2 Fuzzy Daa ons In hs Secon 4, we wll dscuss abou he defnon of T2FD ogeher wh s fuzzfcaon process(alpha-cu operaon), ype-reducon and defuzzfcaon mehods. Defnon 4.1. Gven ha T2FD, whch he hegh of LMF and UMF are

6 (() and h( ) ) h respecvely, hen T2FD called ormal of T2FD(T2FD) ( ) [20]. Ths Def. 4.1 can be llusraed hrough Fg f h( ) < h( ) = 1 µ ( x ) ()) 1 h = (() h (() < ()) = 1 h h UMF ( ), ) Fgure 4.1. The T2FDs. Defnon 4.2. Based on Def. 3.1, le be he se of T2FDs wh where = 0,1,..., n 1. Then s he -cu operaon of normal T2FDs whch s < LMF = h ( < UMF = h ). gven as equaon as follows for ( ) ( ) =,, LMF ( ( ), T1 F( ) = ; ;,, ; ; LMF = ; ; ; ; + ; ;,, CLMF LMF ; ; ; ; + ; ; CLMF (4.1) where LMF and CLMF are alpha values of lower membershp funcon and crsp lower membershp funcon of normal T2FDs respecvely. Ths defnon can be llusraed hrough Fg. 4.2.

7 µ ( x ) = 0.5 = LMF ( ) = 0.5 Fgure 4.2. The alpha-cu operaon owards normal T2FD. However, when LMF < < UMF for -cu operaon of normal T2FDs, hen he Eq. 4.1 become =,, LMF ( ) = = 0.5 = = ; ;0,, 0; ; = ; ;0 ; ;0 + ; ;0,, 0; ; 0; ; + 0; ; (4.2) whch can be llusraed by gven hs followng fgure. µ ( x ) = 0.8 = 0.8 = 0.8 Fgure 4.3. The alpha-cu operaon owards T2FD wh LMF < < UMF. = 0.8

8 Afer he alpha-cu operaon agans T2FDs had been appled, hen he nex sep for obanng crsp ype-2 fuzzy soluon daa pon s he ype-reducon mehod. Type-reducon mehod s a mehod were used o reduce T2FDs o T1FDs afer fuzzfcaon process has been appled for allowng he defuzzfcaon of a ype-1 cases can be appled. There are many ypes of ype-reducon mehod, whch can be referred n [6,7,9,16-19,23,28,29,43]. However, n hs paper, we consruc anoher ype-reducon mehod, whch s known as cenrod mn mehod, whch can be gven va Def Defnon 4.3. Le be a se ( n+ 1) T2FDs, hen ype-reducon mehod of -T2FDs(afer fuzzfcaon), s defned by = =,, ; = 0,1,..., n { } (4.3) where s lef ype-reducon of -cu T2FDs, 1 = + +, s he crsp pon and s rgh 3= 0,..., n 1 ype-reducon of -cu T2FDs, = + + for 3 < <. LMF UMF = 0,..., n However, same goes wh he condon hrough Eq. 4.2, hen he ype-reducon T2FDs for LMF < < UMF s gven by Eq. 4.3 where s lef 1 ype-reducon of -cu T2FDs, = and s 2= 0,..., n 1 rgh ype-reducon of -cu T2FDs, 0 = + +. Then, 2= 0,..., n o fnd he defuzzfy of T2FDs afer ype-reducon mehod has been appled s usng he defuzzfcaon mehod for T1FDs cases, whch defned n Def Defnon 4.4. Le -TR s he ype-reducon mehod afer -cu process had been appled for every T2FDs,. Then named as defuzzfcaon T2FDs for f for every, { } for 0,1,..., = = n (4.4) where for every 1 = <,, >. The process n defuzzfyng 3 = 0

9 T2FDs can be llusraed a Fg µ ( x) µ ( x) µ ( x) = 0.5 T2FD -cu operaon -T2FD µ ( x) µ ( x) Type-reducon process Fgure 4.4. Defuzzfcaon process of T2FD. Fg. 4.4 shows ha he process of defuzzyng he T2FD whch gves he crsp ype-2 fuzzy soluon of daa pons as a fnal resul n defne he complex uncerany daa pons. 5 ormal Type-2 Fuzzy Raonal B-splne Curve Modelng Afer we had fnshed n defnng he complex uncerany daa pons usng T2FST, whch gves T2FD n T2FD form, hen we use he raonal B-splne curve funcon n order o model he T2FDs whch fnally known as normal ype-2 fuzzy raonal B-splne curve(t2frbsc). T2FRBsC wll gve us more comprehended n how normal T2FDs can be modeled and nfluence he raonal curve form. Besdes ha, we can use he defnon of T2FD n defnng he real complex uncerany daa whch we wan o model by usng he exsng curves and surfaces funcon. Therefore, as he hypohecal example of modelng he complex uncerany daa pons, he process of modelng for T2FDs usng Raonal B-splne curve funcon [12,13,27,30-32,42] can be gven n Def. 5.1 as follows. suuuuuur Defnon 5.1. Le RBsCk, n( ) denoe a normal ype-2 fuzzy raonal B-splne of order k (degree, k-1), where k n. Lew for = 1,..., n be he n weghs correspondng o homogeneous normal ype-2 fuzzy daa/conrol pons {,,..., 1 2 n }. Therefore, he T2FRBsC becomes: n suuuuuur k w ( ) 1 = RBsCk, n( ) (5.1) n k w ( ) = = r 1 r r Crsp daa pon Defuzzfy daa pon Defuzzfcaon process TR -T2FD

10 where s he B-splne bass funcon. suuuuuur Afer we defned he RBsCk, n( ) by usng he defnon of T2FST and he oher's suuuuuur relaon defnons, hen we modeled he RBsCk, n( ) ogeher wh he alpha-cu operaon, ype-reducon process and defuzzfcaon mehod n Alg. 1.1 as follows. Algorhm 1.1. Algorhm of fuzzfcaon, ype-reducon and defuzzfcaon processes of T2FRBsC for LMF < < UMF case. Sep 1: Defne four T2FCs, = 1,...,4. Then, he T2FRBsC equaon can be gven by Eq. 5.1 as llusraed by Fg n suuuuuur k w ( ) 1 = RBsCk, n( ), n k = 3; n= 4; = r = 1,..., 4. k w ( ) = = r 1 r r (, w = 1) 2 2 (, w = 1) 4 4 (, w = 1) 1 1 Fgure 5.1. The modelng of T2FRBsC wh w = 1,...,4 = (1,1,3,1). Sep 2: Fuzzfcaon(alpha-cu operaon). For lef(x-axs)/lower(y-axs), + ( ),,0 = 1,...,4 = 1,...,4 0.8 = 1,...,4 where = 1,...,4 = For rgh(x-axs)/upper(y-axs), + ( ) (, w ) = 1,...,4 = 1,...,4 (, w = 3) = 1,...,4 = 1,...,4 0.8 = 1,...,4 where = 1,...,4 = 0,, = 1,...,4 = 1,...,4. (5.2). (5.3) For alpha-cu of dagonal pons, used Eq. 5.2 f ( x, y ) < ( x, y ) or ( x, y ) < ( x, y ) and Eq. 5.3 f ( x, y) < ( x, y) or 3 3

11 ( x, y) < ( x, y). Then, he brand-new T2FDs( -T2FDs) were modeled usng raonal B-splne curve funcon, whch gves -T2FRBsC(Fg. 5.2). n suuuuuur 4 w ( ) 1 = RBsCk, n( ), n k = 3; n= 4; = r = 1,..., 4. k w ( ) = = r 1 r r (, w = 1) 2 2 (, w = 1) 4 4 (, w = 1) 1 1 Fgure T2FRBsC modelng wh w = 1,...,4 = (1,1,3,1). Fg. 5.2 shows he modelng of -T2FRBsC n he cubc form wh he value of s 0.8. Sep 3: Type-reducon process of -T2FRBsC. Based on Eq. 4.3 for LMF < < UMF, hen 1 1 0,...,3 = = + + 0,, =,,. = 1,...,4 = 1,...,4 = 1,...,4 Then, he ype-reducon of -T2FRBsC(TR- -T2FRBsC) model s, n suuuuuur k w ( ) 1 = RBsCk, n( ), n k = 3; n= 4; = r = 1,..., 4. k w ( ) = = r 1 r r (, w = 3) 3 3

12 (, w = 1) 2 2 (, w = 1) 4 4 (, w = 1) 1 1 Fgure 5.3. TR- -T2FRBsC modelng wh w = 1,...,4 = (1,1,3,1). Fg. 5.3 shows ha he modelng of TR- -T2FRBsC. The TR- -T2FRBsC also becomes he ype-1 fuzzy curve form whch allows us o defuzzfy he TR- -T2FRBsC o become crsp T2FRBs soluon curve(d-tr- -T2FRBsC). The mehod used o defuzzfy he TR- -T2FRBsC s he cenrod mn mehod [2-4,34,39,37]. Sep 4: Defuzzfcaon of TR- -T2FRBsC. (, w = 3) = + + = 1,...,4 3= 0,...,3 = = 1,...,4. = 1,...,4 Then, he crsp T2FRBs soluon curve model s gven by Fg RBsCk, n ( ) n k w ( ) 1 = n k w ( ) r 1 r r = =, k = 3; n= 4; = r = 1,..., 4.

13 ((, ), w = 1) ((, ), w = 1) ((, ), w = 1) Fgure 5.4. Crsp T2FRBs soluon curve, RBsCk, n( ) marked by blue pons ogeher wh crsp raonal B-plne curve marked by red pons. Fg. 5.4 shows ha he modelng of RBsCk, n( ) along wh he crsp raonal B-splne curve. There are some errors beween he crsp T2FDs soluon and daa pons. Ths s happened because he T2FDs are no equally he same, meanng ha he lengh beween he lef T2FDs and crsp conrol pons are no equal hroughou he lengh beween he rgh T2FDs and crsp conrol pons. 6 Concluson The consrucon of T2FDs o deal wh complex uncerany daa se had been consruced by usng he fundamenal of T2FST and beng modeled by usng raonal B-splne curve funcon. The T2FDs n he normal form(t2fds) was defned, where hs defnon gves us advanage n dealng he varey of T2FDs. Ths mehod can be appled n dealng wh complex daa uncerany, whch exss n real-lfe applcaon. Currenly, hs mehod s exended for surface desgn by usng ensor produc echnque for surface modelng. Acknowledgemen ((, ), w = 3) The auhors would lke o hank Research Managemen and Innovaon Cenre (RMIC) of Unvers Malaysa Terengganu and Mnsry of Hgher Educaon (MOHE) Malaysa for fundng(frgs, vo59244) and provdng he facles o carry ou hs research.

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