pped Mahemaca Scence, Vo. 7, 203, no. 45, 2239 2252 HIKRI Ld, www.m-hkar.com Generazed orma Type-2 Trangar Fzzy mber bd. Faah Wahab Deparmen of Mahemac, Facy of Scence and Technoogy, Unver Maaya Terenggan, Maaya. faah@m.ed.my Rozam Zakara Deparmen of Mahemac, Facy of Scence and Technoogy, Unver Maaya Terenggan, Maaya. rozam_z@yahoo.com Copyrgh 203 bd. Faah Wahab and Rozam Zakara. Th an open acce arce drbed nder he Creave Common rbon Lcene, whch perm nrerced e, drbon, and reprodcon n any medm, provded he orgna work propery ced. brac Here, we preen for heorem nvovng norma ype-2 rangar fzzy nmber (T2TF. Keyword: Type-2 rangar fzzy nmber, pha-c, Type-redcon, Defzzfcaon Inrodcon Type-2 fzzy nmber (T2F concep wa nrodced a he exenon of ype- fzzy nmber (TF [2,5] concep n deang he probem of defnng he compex ncerany daa n rea daa form. Th T2F defned by he ype-2 fzzy e (T2FS heory whch wa nrodced by Zadeh [3] n order o ove he compex ncerany probem of rea daa e. Therefore, he defnon of T2F, norma T2F, apha-c operaon, ype-redcon and defzzfcaon proce of norma T2F are gven a foow. Defnon. T2F broady defned a a T2FS ha ha a nmerca doman. n nerva T2FS defned ng he foowng for conran, where
2240 bd. Faah Wahab and Rozam Zakara a b c d = {[, ],[, ]}, [ 0,], a b, b, c, d R (Fg. []:. a b c d 2. [ a, d ] and [ b, c ] generae a fncon ha convex and [ a, d ] generae a fncon norma. 2 2 3., 2 [0,]:( 2 > ( a, c a, c, 2 2 2 2 b, d b, d, for c b. 4. If he maxmm of he memberhp fncon generaed by [ b, c ] he m m eve m, ha, [ b, c ], hen m m = = b, c a, d. 2 a b c d x 0 0 a b c d Fgre. Defnon of an nerva T2F. x Defnon 2. Gven ha T2F, whch he hegh of ower memberhp fncon(lmf and pper memberhp fncon(umf are h ( ( and h( repecvey, hen T2F caed orma of T2F(T2F f h( ( < h( = [4]. Th Def. 2 can be raed hrogh Fg. 2. μ ( x 0.8 0.6 ( = h ( ( h ( ( < h( = h 0.4 0.2 0 0 2 4 6 8 0 2 P (, LMF ( TF( P (, UMF Fgre 2. The T2F.
Generazed norma ype-2 rangar fzzy nmber 224 Defnon 3. Baed on Def. 2, e be he e of T2F n rangar form wh where = 0,,..., n. Then he apha-c operaon of T2TF whch gven a eqaon a foow [4]. =,, P = ; ;,, ; ; LMF = ; ; ; ; + ; ;,, CLMF LMF ; ; ; ; + ; ; CLMF ( where LMF and CLMF are apha vae of ower memberhp fncon and crp ower memberhp fncon of T2TF repecvey. Th defnon can be raed hrogh Fg. 3. μ ( x.2 0.8 0.6 0.4 0.2 0 0 2 4 6 8 0 2 Fgre 3. The apha-c operaon oward T2TF. However, when LMF < < UMF for -c operaon of T2TF, hen he Eq. become =,, P = = 0.5 = 0.5 ( LMF = 0.5 ; ;,, ; ; ( LMF = 0.5 P 0.5 = P 0.5 =
2242 bd. Faah Wahab and Rozam Zakara = ; ; ; ;0 + ; ;,, ; ; 0; ; + ; ; (2 whch can be raed by gven h foowng fgre. μ ( x 0. 0. 0. 0. Fgre 4. The apha-c operaon oward T2TF wh LMF < < UMF. Defnon 4. Le be a e of ( n + T2TF, hen ype-redcon mehod of -T2TF(afer fzzfcaon, defned [4] by = { =,, ; = 0,,..., n } (3 where are ef ype-redcon of apha-c T2TF, = + +, he crp pon of T2TF and 3 = 0,..., n rgh ype-redcon of apha-c T2TF, = + + P P P. 3 = 0,..., n Defnon 5. Le -TR he ype-redcon mehod afer -c proce had been apped for every T2TF,. Then named a defzzfcaon T2TF for f for every [4], { } for 0,,..., 0 0 2 4 6 8 = 0.8 = 0.8 = 0.8 = 0.8 = = n where for every =,, < >. The proce n defzzfyng he 3 = 0
Generazed norma ype-2 rangar fzzy nmber 2243 T2TF can be raed a Fg. 5. μ ( x μ ( x μ( x = 0.5 T2TF -c operaon -T2TF μ( x μ( x Type-redcon proce Crp pon Defzzfy pon Defzzfcaon proce Fgre 5. Defzzfcaon proce of T2TF. TR -T2TF 2 Re Theorem 2.. Le be a TT2F whch cenred a c wh < ϕ, εγ, > and < η, φλ, > are ef and rgh nerva(nerva of fooprn of repecvey. If and are he -c of where < wh < < whch and are ower and pper -c of, hen (, (., Proof. Le he memberhp fncon of gven a c x, f ( ϕεγ,, x c ( ϕεγ,, x c x ( =, f c x ( η, φλ, (4 ( ηφλ,, 0, oherwe where ϕ ε γ and η φ λ. Then, he nerva of < c ( ϕ, εγ,, c, c + ( ηφλ,, >. From Eq. 4, f he ype-2 fzzy nerva wa obaned by -c operaon, hen he nerva of acheved whch gven a =< r,, r > (,, c and =< r,, r > (,, (,, c wh (,, and r and,, r,, and r,, are ef and rgh fooprn of r,,
2244 bd. Faah Wahab and Rozam Zakara -c of where < for a, (0,]. For, ( c ( ϕεγ,, r ( ( ϕεγ,, r c (,, r = (,, =, c, c ( c ( ϕεγ,, ( c+ ( ηφλ,, ( c+ ( ηφλ,, r r (,, r = ( =,, ( c+ ( ηφλ,, c r r ( ( ϕεγ,, r c = [ c ( c ( ϕεγ,, ](,, ( (,,,, (,, = + c ϕ ε γ c r r ( + ( ηφλ,, c r = [( c+ ( ηφλ,, c]( =,, + ( + ( ηφλ,,. (,, c For, ( c ( ϕεγ,, r ( ( ϕεγ,, r c (,, r = (,, =, c, c ( c ( ϕεγ,, ( c+ ( ηφλ,, ( c+ ( ηφλ,, r r (,, r = ( =,, ( c+ ( ηφλ,, c r r ( c ( ϕεγ,, r = [ c ( c ( ϕεγ,, ](,, ( (,,,, (,, = + c ϕεγ c r r ( c+ ( ηφλ,, r = [( c+ ( ηφλ,, c]( =,, + ( + ( ηφλ,,. (,, c Snce <, hen ( c ( ϕεγ,, r < ( c ( ϕεγ,, r, c, (,, (,, ( c+ ( ηφλ,, r < ( + ( ηφλ,, c r (,, (,, r r = ([ c ( c ( ϕεγ,, ](,, = + ( c ( ϕεγ,, < ([ c ( c r r ( ϕεγ,, ](,, = + ( c ( ϕεγ,,, c,( [( c+ ( ηφλ,, r r c]( =,, + ( c+ ( η, φ, λ < ( [( c+ ( η, φ, λ
Generazed norma ype-2 rangar fzzy nmber 2245 r r c]( =,, + ( c+ ( η, φ, λ r r r r r r r r = (,, = < (,, =, c,( =,, < ( =,, Then, r < r,, < (,, (,, c (,, (,,. Therefore, < r,, > < r,, (,, c r r > (,, (,, c (,, (, (, μ ( x 0 c ϕ c ε c γ c c η c φ c λ x (, (, Fgre 6. < (, (,. Theorem 2... If and are he -c of where < wh < < whch and are ower and pper -c of, hen. Proof. For,
2246 bd. Faah Wahab and Rozam Zakara ( c ( ϕε,,0 ( (,,0 c ϕε (,,0 = (,,0, c, c ( c ( ϕε,,0 ( c+ (0, φλ, ( c+ (0, φλ, ( c+ (0, φλ, c (0,, (0,, = (0,, r ( c ( ϕε,,0 = [ c ( c ( ϕε,,0](,,0 ( (,,0,, (,,0 + c ϕε c ( c+ (0, φλ, = [( c+ (0, φλ, c](0,, + ( c+ (0, φλ,. For, ( c ( ϕε,,0 ( c ( ϕε,,0 (,,0 = (,,0, c, c ( c ( ϕε,,0 ( c+ (0, φλ, ( c+ (0, φλ, ( c+ (0, φλ, c (,,0 (0,, (0,, = (0,, ( c ( ϕε,,0 = [ c ( c ( ϕε,,0](,,0 + ( c ( ϕε,,0, c, ( c+ (0, φλ, = [( c+ (0, φλ, c](0,, + ( c+ (0, φλ,. Snce <, hen ( c ( ϕε,,0 < ( c ( ϕε,,0, c, (,,0 (,,0 ( c+ (0, φλ, < ( + (0, φλ, c (0,, (0,, = ([ c ( c ( ϕε,,0](,,0 + ( c ( ϕε,,0 < ([ c ( c ( ϕε,,0](,,0 + ( c ( ϕε,,0, c,( [( c+ (0, φλ, c](0,, + ( c+ (0, φλ, < ( [( c+ (0, φλ, c](0,, + ( c+ (0, φλ, = (,,0 < (,,0, c,(0,, < (0,, Then, <,, < (,,0 (,,0 c (0,, (0,,. Therefore, <,, > <,, (,,0 c > (0,, (,,0 c (0,, where, (,
Generazed norma ype-2 rangar fzzy nmber 2247 μ ( x 0 c ϕ c ε c γ c c η c φ c λ x Fgre 7. <. Theorem 2..2. If and are he -c of where < wh < < whch and are ower and pper -c of, hen. (, Proof. For, ( c ( ϕεγ,, r ( ( ϕεγ,, r c (,, r = (,, =, c, c ( c ( ϕεγ,, ( c+ ( ηφλ,, ( c+ ( ηφλ,, r r (,, r = ( =,, ( c+ ( ηφλ,, c r r ( ( ϕεγ,, r c = [ c ( c ( ϕεγ,, ](,, ( (,,,, (,, = + c ϕ ε γ c r r ( + ( ηφλ,, c r = [( c+ ( ηφλ,, c]( =,, + ( + ( ηφλ,,. (,, c For,
2248 bd. Faah Wahab and Rozam Zakara ( c ( ϕεγ,, ( c ( ϕεγ,, (,,0 = (,,0, c, c ( c ( ϕεγ,, ( c+ ( ηφλ,, ( c+ ( ηφλ,, ( c+ ( ηφλ,, c (,,0 (0,, = (0,, ( c ( ϕεγ,, = [ c ( c ( ϕεγ,, ](,,0 + ( c ( ϕεγ,,, c, ( + ( ηφλ,, [( ( ηφλ,, ](0, r, r = c+ c + ( c+ ( ηφλ,,. c (0,, Snce <, hen ( c ( ϕεγ,, < ( c ( ϕε,,0, c, r (,, (,,0 ( c+ ( ηφλ,, < ( c+ (0, φλ, r (,, (0,, r r = ([ c ( c ( ϕεγ,, ](,, = + ( c ( ϕεγ,, < ([ c ( c ( ϕεγ,, ](,,0 r r + ( c ( ϕεγ,,, c,( [( c+ ( ηφλ,, c]( =,, + ( c+ ( ηφλ,, < ( [( c+ ( ηφλ,, c](0,, + ( c+ ( ηφλ,, r r r r = (,, = < (,,0, c,( =,, < (0,, Then, r <,, < (,, (,,0 c r (,, (0,,. Therefore, <,, > < r,, (,,0 c r > (0,, (,, c (,, (, where (0,] and (,]
Generazed norma ype-2 rangar fzzy nmber 2249 μ ( x 0 c ϕ c ε c γ c c η c φ c λ x Fgre 8. < (,. Theorem 2.2. Baed on he defnon of defzzfcaon for T2TF, e be a repreenaon of -c operaon of T2TF, and ( TR be ype-redcon of, whch gve = = ( ( ϕεγ,,,, = ( + ( ηφλ,, and = = ( ( ϕεγ,,,, = ( + ( ηφλ,, ( TR ( TR ( TR ( TR ( TR where and are ef and rgh fooprn of T2TF afer -c operaon wa apped and ( TR and ( TR are ef and rgh fooprn -T2TF afer ype-redcon ha been apped, a crp pon and ( ϕ, εγ, are ef-ef, ef and rgh-ef engh and ( η, φλ, are ef-rgh, rgh, rgh-rgh engh from repecvey wh γ < ε < ϕ and λ < φ < η. If ( η, φλ, < ( ϕεγ,, or ( ϕ, εγ, < ( ηφλ,,, hen crp ype-2 fzzy oon on ef ogh of repecvey. Proof. Gven ha = = ( ( ϕεγ,,,, = ( + ( ηφλ,,. Then, we obaned (, Cae : For < < where and are ower and pper memberhp fncon of, hen
2250 bd. Faah Wahab and Rozam Zakara ( ( ϕεγ,, ( ( ϕεγ,, ( + ( ηφλ,, ( + ( ηφλ,,,, ( ( ϕεγ,, ( + ( ηφλ,, = ( ( ϕεγ,, = ( ( ( ϕεγ,, (,, + ( ( ϕεγ,,,, ( + ( η, φλ, = (( + ( ηφλ,, (,, + ( + ( ηφλ,, = ( ( ϕεγ,,,,( + ( η, φλ,. For ( η, φλ, < ( ϕεγ,,, hen = (( + ( ηφλ,, (,, + ( + ( ηφλ,, < ( ( ( ϕεγ,, (,, + ( ( ϕεγ,, = (,, ( ηφλ,, (,, + (,, + + ( ηφλ,, < (,, (,, + ( ϕεγ,, (,, + ( ϕεγ,, = ( ηφλ,, (,, + ( ηφλ,, < ( ϕεγ,, (,, ( ϕεγ,, = ( ηφλ,, ((,, < ( ϕεγ,, ((,, = ( ηφλ,, < ( ϕεγ,,. For ( ϕ, εγ, < ( ηφλ,,, hen = ( ( ( ϕ, εγ, (,, + ( ( ϕεγ,, < (( + ( ηφλ,, (,, + ( + ( ηφλ,, = (,, (,, + ( ϕεγ,, (,, + ( ϕεγ,, < (,, ( ηφλ,, (,, + (,, + + ( ηφλ,, = ( ϕ, εγ, (,, ( ϕεγ,, < ( ηφλ,, (,, + ( ηφλ,, = ( ϕεγ,, ( (,, < ( ηφλ,, ( (,, = ( ϕ, εγ, < ( ηφλ,,. The ype-redcon proce of, ( TR gven a
Generazed norma ype-2 rangar fzzy nmber 225 = = ( ( ϕεγ,,,, = ( + ( ηφλ,, ( TR ( TR ( TR ( TR ( TR = ( ϕ + ( ε + ( γ ( + η + ( + φ + ( + λ,,. 3 3 Then, he defzzfcaon proce of ( can be gven a foow TR ( TR ( ϕ + ( ε + ( γ ( + η + ( + φ + ( + λ + + 3 3 =. 3 Therefore, f crp ype-2 fzzy oon wa obaned a he ef de of, hen whch ( + ( ηφλ,, < ( ( ϕεγ,, wh ( η, φλ, < ( ϕεγ,, ( TR < and f crp ype-2 fzzy oon a he rgh de of, hen < ( TR whch ( ( ϕ, εγ, < ( + ( ηφλ,, wh ( ϕ, εγ, < ( ηφλ,,. Cae 2: For < <, hen ( ( ϕε,,0 ( ( ϕε,,0 ( + (0, φλ, ( + (0, φλ,,, ( ( ϕε,,0 ( + (0, φλ, = ( ( ϕε,,0 = ( ( ( ϕε,,0(,,0 + ( ( ϕε,,0,, ( ( + (0, φ, λ = ( + (0, φ, λ (0,, + ( + (0, φ, λ = ( ( ϕε,,0,,( + (0, φλ,. For ( η, φλ, < ( ϕεγ,,, hen ( = ( + (0, φλ, (0,, + ( + (0, φλ, < ( ( ( ϕε,,0(,,0 + ( ( ϕε,,0 = (0,, (0, φλ, (0,, + (0,, + + (0, φλ, < (,,0 (,,0 + ( ϕε,,0(,,0 + ( ϕε,,0 = (0, φλ, (0,, + (0, φλ, < ( ϕε,,0(,,0 ( ϕε,,0 = (0, φλ, ((0,, < ( ϕε,,0((,,0 = (0, φλ, < ( ϕε,,0. The ype-redcon proce of, ( gven a TR
2252 bd. Faah Wahab and Rozam Zakara = = ( ( ϕε,,0,, = ( + ( ηφ,,0 ( TR ( TR ( TR ( TR ( TR = ( ϕ + ( ε + 0 0 + ( + φ + ( + λ,,. 2 2 Then, he defzzfcaon proce of ( TR can be gven a foow ( ϕ + ( ε + 0 0 + ( + φ + ( + λ + + 2 2 ( TR =. 3 Therefore, f crp ype-2 fzzy oon wa obaned a he ef de of, hen whch ( + (0, φλ, < ( ( ϕε,,0 wh (0, φ, λ < ( ϕ, ε,0 ( TR < and f crp ype-2 fzzy oon a he rgh de of, hen < ( TR whch ( ( ϕ, ε,0 < ( + (0, φ, λ wh ( ϕ, ε,0 < (0, φ, λ cknowedgemen The ahor wod ke o hank Reearch Managemen and Innovaon Cenre (RMIC of Unver Maaya Terenggan and Mnry of Hgher Edcaon (MOHE Maaya for fndng(frgs, vo59244 and provdng he face o carry o h reearch. Reference [] J.R. gero,. Varga. (2007. Cacang Fncon of Inerva Type-2 Fzzy mber for Fa Crren nay. IEEE Tranacon on Fzzy Syem, 5(, 3-40. [2] D. Dbo, H. Prade. (980. Fzzy Se and Syem: Theory and ppcaon. ew York: cademc Pre. [3] L.. Zadeh. (975. The concep of a ngc varabe and appcaon o approxmae reaonng-par I-II-III Informaon Scence, 8, 8, 9, 99-249, 30-357, 43-80. [4] R. Zakara,.F. Wahab, R.U. Gobhaaan. (203. orma Type-2 Fzzy Raona B-pne Crve. Inernaona Jorna of Mahemaca nay, 7(6, 789-806. [5] H.-J. Zmmermann. (985. Fzzy Se Theory and I ppcaon. US: Kwer cademc. Receved: Febrary, 203