Generalized Normal Type-2. Triangular Fuzzy Number

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1 pped Mahemaca Scence, Vo. 7, 203, no. 45, HIKRI Ld, Generazed orma Type-2 Trangar Fzzy mber bd. Faah Wahab Deparmen of Mahemac, Facy of Scence and Technoogy, Unver Maaya Terenggan, Maaya. Rozam Zakara Deparmen of Mahemac, Facy of Scence and Technoogy, Unver Maaya Terenggan, Maaya. 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

2 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 [0,]:( 2 > ( a, c a, c, 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 ( = h ( ( h ( ( < h( = h P (, LMF ( TF( P (, UMF Fgre 2. The T2F.

3 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 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 =

4 2242 bd. Faah Wahab and Rozam Zakara = ; ; ; ;0 + ; ;,, ; ; 0; ; + ; ; (2 whch can be raed by gven h foowng fgre. μ ( x 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,,..., = 0.8 = 0.8 = 0.8 = 0.8 = = n where for every =,, < >. The proce n defzzfyng he 3 = 0

5 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,,

6 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

7 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,

8 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, (,

9 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 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,

10 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 (,]

11 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

12 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

13 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 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

14 2252 bd. Faah Wahab and Rozam Zakara = = ( ( ϕε,,0,, = ( + ( ηφ,,0 ( TR ( TR ( TR ( TR ( TR = ( ϕ + ( ε ( + φ + ( + λ,,. 2 2 Then, he defzzfcaon proce of ( TR can be gven a foow ( ϕ + ( ε ( + φ + ( + λ ( 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(, [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, , , [4] R. Zakara,.F. Wahab, R.U. Gobhaaan. (203. orma Type-2 Fzzy Raona B-pne Crve. Inernaona Jorna of Mahemaca nay, 7(6, [5] H.-J. Zmmermann. (985. Fzzy Se Theory and I ppcaon. US: Kwer cademc. Receved: Febrary, 203