A Generalization of Generalized Triangular Fuzzy Sets

International Journal of Mathematical Analysis Vol, 207, no 9, 433-443 HIKARI Ltd, wwwm-hikaricom https://doiorg/02988/ijma2077350 A Generalization of Generalized Triangular Fuzzy Sets Chang Il Kim Department of Mathematics Education Dankook University 26 Jukjeon-dong, Yongin-si, Kyunggi-do 6890, Korea Yong Sik Yun* Department of Mathematics Research Institute for Basic Sciences Jeju National University Jeju 63243, Korea Copyright c 207 Chang Il Kim Yong Sik Yun This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited Abstract A generalized triangular fuzzy set is a triangular fuzzy set which does not have a maximum value We calculated Zadeh s max-min composition operator for two generalized triangular fuzzy sets By using parametric operations between two regions valued α-cuts, we generated the generalized triangular fuzzy sets on R to R 2 calculated the parametric operations for two generalized 2-dimensional triangular fuzzy sets We show that the parametric operations for two generalized triangular fuzzy sets defined on R 2 are a generalization of Zadeh s max-min composition operations for two generalized triangular fuzzy sets defined on R Mathematics Subject Classification: 47N30, 60D05 *Corresponding author *This research was supported by the 206 scientific promotion program funded by Jeju National University

434 Chang Il Kim Yong Sik Yun Keywords: generalization, a generalized 2-dimensional triangular fuzzy set Introduction The membership function of triangular fuzzy number consists of monotone increasing decreasing functions which have a maximum value The extended algebraic operations between two triangular fuzzy numbers are wellknown A generalized triangular fuzzy set is a triangular fuzzy set which does not have a maximum value We calculated Zadeh s max-min composition operator for two generalized triangular fuzzy sets([4]) In [], we generated the triangular fuzzy numbers on R to R 2 By defining parametric operations between two regions valued α-cuts, we got the parametric operations for two triangular fuzzy numbers defined on R 2 The results for the parametric operations are the generalization of Zadeh s extended algebraic operations We proved that the results for the parametric operations became the generalization of Zadeh s extended algebraic operations Moreover, we generated the generalized triangular fuzzy sets on R to R 2 calculated the parametric operations for two generalized 2-dimensional triangular fuzzy sets([3]) In this paper, we will show that the parametric operations for two generalized triangular fuzzy sets defined on R 2 are a generalization of Zadeh s max-min composition operations for two generalized triangular fuzzy sets defined on R 2 Zadeh s max-min composition operations for generalized triangular fuzzy sets on R Definition 2 An α-cut of the fuzzy number A is defined by A α = {x R µ A (x) α if α (0, ] A α = cl{x R µ A (x) > α if α = 0 For α (0, ), the set A α = {x X µ A (x) = α is said to be the α-set of the fuzzy set A A 0 is the boundary of {x R µ A (x) > α A = A In the calculations between two fuzzy numbers, the concept of α-cut is very important Furthermore, some operations between α-cuts are useful α-set plays a crucial role in a 2-dimensional case Definition 22 ([5]) The extended addition A(+)B, extended subtraction A( )B, extended multiplication A( )B extended division A(/)B are fuzzy sets with membership functions as follows For all x A y B, µ A( )B (z) = sup min{µ A (x), µ B (y), = +,,, / z=x y

A generalization of a generalized triangular fuzzy sets 435 Definition 23 A fuzzy set A which has a membership function 0, x < a, a 4 x, x a a µ A (x) = 2 a, a x < a 2, c, a 2 x < a 3, a 4 x a 4 a 3, a 3 x < a 4, where a i R, i =, 2, 3, 4 0 < c, is called a generalized trapezoidal fuzzy set It is denoted by A = (a, a 2, c, a 3, a 4 ) We generalize the triangular fuzzy number A generalized triangular fuzzy set is symmetric may not have value Definition 24 A generalized triangular fuzzy set is a symmetric fuzzy set A which has a membership function 0, x < a, a 2 x, 2c(x a µ A (x) = ) a 2 a, a x < a +a 2, 2 2c(x a 2 ) a a 2 a, +a 2 x < a 2 2, where a, a 2 R 0 < c The triangular fuzzy set generalized above is denoted by A = ((a, c, a 2 )) The following Theorem 25 is a correction of Theorem 32 in [4] Theorem 25 For two generalized triangular fuzzy sets A = ((m, c, m 2 )) B = ((n, c 2, n 2 )), if c c 2 µ B (x) c in [k, k 2 ], we have the followings () A(+)B = (m + n, (m 2 + m 2 ) + k, c, (m 2 + m 2 ) + k 2, m 2 + n 2 ), ie, A(+)B is a generalized trapezoidal fuzzy set (2) A( )B = (m n 2, (m 2 + m 2 ) k 2, c, (m 2 + m 2 ) k, m 2 n ), ie, A( )B is a generalized trapezoidal fuzzy set (3) A( )B is a fuzzy set on (m n, m 2 n 2 ), but needs not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set The membership function of A( )B is 0, x < m n, m 2 n 2 x, 2pq( pn qm + (pn + qm 4pq(m n x) ), m n x < µ A( )B (x) = (m 2 + m 2 )k, c, (m 2 + m 2 )k x < (m 2 + m 2 )k 2, 2pq( pn2 + qm 2 (pn 2 + qm 2 4pq(m 2 n 2 x) ), (m 2 + m 2 )k 2 x < m 2 n 2,

436 Chang Il Kim Yong Sik Yun where p = m 2 m 2c q = n 2 n 2c 2 (4) A(/)B is a fuzzy set on ( m n 2, m 2 n ), but needs not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set The membership function of A(/)B is µ A(/)B (x) = 0, x < m n 2, m 2 n x, 2c c 2 (n 2 x m ) c 2 (m 2 m )+c (n 2 n )x m n 2 x < m +m 2 2k 2, c, m +m 2 2k 2 x < m +m 2 2k, 2c c 2 (n x m 2 ) c 2 (m 2 m )+c (n 2 n )x, m +m 2 2k x < m 2 3 A generalization of generalized triangular fuzzy sets We defined the generalized 2-dimensional triangular fuzzy numbers on R 2 as a generalization of generalized triangular fuzzy numbers on R the parametric operations between two generalized 2-dimensional triangular fuzzy numbers For that, we have to calculate operations between α-cuts in R The α-cuts are intervals in R but in R 2 the α-cuts are regions, which makes the existing method of calculations between α-cuts unusable We interpreted the existing method from a different perspective apply the method to the region valued α-cuts on R 2 In this section, we will show that the parametric operations for two generalized triangular fuzzy sets defined on R 2 is a generalization of Zadeh s max-min composition operations for two generalized triangular fuzzy sets defined on R For that, we have to prove that the intersections of the results on R 2 vertical plane are as same as that on R Definition 3 A fuzzy set A with a membership function µ A (x, y) = { h (x x a 2 + (y y, b 2 b 2 (x x + a 2 (y y a 2 b 2 h 2, 0, otherwise, where a, b > 0 0 < h < is called the generalized 2-dimensional triangular fuzzy number denoted by ((a, x, h, b, y ) Note that µ A (x, y) is a cone The intersections of µ A (x, y) the horizontal planes z = α (0 < α < h) are ellipses The intersections of µ A (x, y) the vertical planes y y = k(x x ) (k R) are symmetric triangular fuzzy numbers in those planes If a = b, ellipses become circles The α-cut A α of a generalized 2-dimensional triangular fuzzy number A = (a, x, h, b, y is an interior of ellipse in an xy-plane including the boundary n

A generalization of a generalized triangular fuzzy sets 437 A α = {(x, y) R 2 b 2 (x x + a 2 (y y a 2 b 2 (h α { ( = (x, y) R 2 x x ( y y + a(h α) b(h α) Definition 32 A 2-dimensional fuzzy number A defined on R 2 is called convex fuzzy number if for all α (0, ), the α-cuts are convex subsets in R 2 A α = {(x, y) R 2 µ A (x, y) α Theorem 33 ([]) Let A be a convex fuzzy number defined on R 2 A α = {(x, y) R 2 µ A (x, y) = α be the α-set of A Then for all α (0, ), there exist piecewise continuous functions f α (t) f2 α (t) defined on [0, 2π] such that A α = {(f α (t), f α 2 (t)) R 2 0 t 2π If A is a continuous convex fuzzy number defined on R 2, then the α-set A α is a closed circular convex subset in R 2 Corollary 34 ([]) Let A be a continuous convex fuzzy number defined on R 2 A α = {(x, y) R 2 µ A (x, y) = α be the α-set of A Then for all α (0, ), there exist continuous functions f α (t) f α 2 (t) defined on [0, 2π] such that A α = {(f α (t), f α 2 (t)) R 2 0 t 2π Definition 35 ([]) Let A B be convex fuzzy numbers defined on R 2 A α = {(x, y) R 2 µ A (x, y) = α = {(f α (t), f α 2 (t)) R 2 0 t 2π, B α = {(x, y) R 2 µ B (x, y) = α = {(g α (t), g α 2 (t)) R 2 0 t 2π be the α-sets of A B, respectively For α (0, ), the parametric addition, parametric subtraction, parametric multiplication parametric division are fuzzy numbers that have their α-sets as follows () parametric addition A(+) p B : (A(+) p B) α = {(f α (t) + g α (t), f α 2 (t) + g α 2 (t)) R 2 0 t 2π

438 Chang Il Kim Yong Sik Yun (2) parametric subtraction A( ) p B : (A( ) p B) α = {(x α (t), y α (t)) R 2 0 t 2π, where x α (t) = y α (t) = { f α (t) g α (t + π), f α (t) g α (t π), { f α 2 (t) g α 2 (t + π), f α 2 (t) g α 2 (t π), if 0 t π if π t 2π if 0 t π if π t 2π (3) parametric multiplication A( ) p B : (A( ) p B) α = {(f α (t) g α (t), f2 α (t) g2 α (t)) R 2 0 t 2π (4) parametric division A(/) p B : (A(/) p B) α = {(x α (t), y α (t)) R 2 0 t 2π, where x α (t) = f α (t) g α (t + π) (0 t π), x α (t) = f α (t) g α (t π) (π t 2π) y α (t) = f α 2 (t) g α 2 (t + π) (0 t π), y α (t) = f α 2 (t) g α 2 (t π) (π t 2π) For α = 0 α =, (A( ) p B) 0 = lim α 0 +(A( ) p B) α (A( ) p B) = lim α (A( ) p B) α, where = +,,, / For 0 < h < h 2, let A = ((a, x, h, b, y ) B = ((a 2, x 2, h 2, b 2, y 2 ) be two generalized 2-dimensional triangular fuzzy numbers If 0 α < h, (A( ) p B) α can be defined as same as Definition 35 If α = h, (A( ) p B) h = lim (A( ) p B) α, = +,,, / α h Then (A( ) p B) h becomes ellipse not a point If h < α h 2, by Zadeh s max-min principle operations, we have to define (A( ) p B) α = (A( ) p B) h, = +,,, /

A generalization of a generalized triangular fuzzy sets 439 Theorem 36 ([3]) Let A = ((a, x, h, b, y ) B = ((a 2, x 2, h 2, b 2, y 2 ) be two generalized 2-dimensional triangular fuzzy numbers If 0 < h < h 2, then we have the following () For 0 < α < h, the α-set of A(+) p B is ( (A(+) p B) α = {(x, y) R 2 x x x 2 a (h α) + a 2 (h 2 α) ( + (2) For 0 < α < h, the α-set of A( ) p B is y y y 2 b (h α) + b 2 (h 2 α) ( (A( ) p B) α = {(x, y) R 2 x x + x 2 a (h α) + a 2 (h 2 α) ( + (3) (A( ) p B) α = {(x α (t), y α (t)) 0 t 2π, where y y + y 2 b (h α) + b 2 (h 2 α) x α (t) = x x 2 + (x a 2 (h 2 α) + x 2 a (h α)) cos t + a a 2 (h α)(h 2 α) cos 2 t, 0 < α < h, y α (t) = y y 2 + (y b 2 (h 2 α) + y 2 b (h α)) sin t + b b 2 (h α)(h 2 α) sin 2 t, 0 < α < h (4) (A(/) p B) α = {(x α (t), y α (t)) 0 t 2π, where = = x α (t) = x + a (h α) cos t x 2 a 2 (h 2 α) cos t, y α(t) = y + b (h α) sin t y 2 b 2 (h 2 α) sin t, 0 < α < h Theorem 37 For = +,,, /, let µ A( )B (x, y) be the results in Theorem 36 µ A( )B (x) be the results in Theorem 25 Let µ A( )B (x, 0) be the fuzzy sets on xz-plane such that µ A( )B (x, 0) {xz plane = µ A( )B (x, 0) Then we have µ A( )B (x, 0) = µ A( )B(x) P roof In Theorem 25, two generalized triangular fuzzy sets A B are symmetric Thus we consider only symmetric case in Theorem 36 Let a = b a 2 = b 2 in Theorem 36 Since A = ((a, x, h, a, y ), µ A (x, y) = 0 a 2 (x x + a 2 y 2 = h 2 a 4

440 Chang Il Kim Yong Sik Yun If y = 0, we have x = x ± h a Note that these x h a x + h a are as same as m m 2 in Theorem 25, respectively Similarly, for µ B (x, y), x 2 h 2 a 2 x 2 + h 2 a 2 are as same as n n 2 in Theorem 4, respectively For solving k, k 2, put µ B (x, y) = h 2 (x x 2 a 2 2 + y2 a 2 2 = h If y = 0, we have x = x 2 ± a 2 (h 2 h ) Note that these x 2 a 2 (h 2 h ) x 2 +a 2 (h 2 h ) are as same as k k 2 in Theorem 25, respectively Clearly, c = h c 2 = h 2 To sum up, we have m = x h a, n = x 2 h 2 a 2, k = x 2 a 2 (h 2 h ), c = h, m 2 = x + h a, n 2 = x 2 + h 2 a 2, k 2 = x 2 + a 2 (h 2 h ), c 2 = h 2 () In Theorem 36, for 0 < α < h, the α-set of A(+) p B is ( (A(+) p B) α = {(x, y) R 2 x x x 2 a (h α) + a 2 (h 2 α) ( + If y = 0, the 0-set of A(+) p B is y y y 2 b (h α) + b 2 (h 2 α) = { ( lim = (x, 0) R 2 x x x 2 = α 0 +(A(+)B)α a h + a 2 h 2 Thus the 0-cut of µ A(+)B (x, 0) is [x + x 2 (a h + a 2 h 2 ), x + x 2 + (a h + a 2 h 2 )] Applying Theorem 25, we have x + x 2 (a h + a 2 h 2 ) = m + n x + x 2 + (a h + a 2 h 2 ) = m 2 + n 2 If y = 0, the h -set of A(+) p B is { lim (A(+)B) α = α h (x, 0) R 2 ( x x x 2 a 2 (h 2 h ) = Thus the h -cut of µ A(+)B (x, 0) is [x + x 2 a 2 (h 2 h ), x + x 2 + a 2 (h 2 h )] Applying Theorem 25, we have x + x 2 a 2 (h 2 h ) = 2 (m + m 2 ) + k x + x 2 + a 2 (h 2 h ) = 2 (m + m 2 ) + k 2 Since A(+)B is a generalized trapezoidal fuzzy set, we have µ A(+)B (x, 0) = µ A(+)B(x) (2) In Theorem 36, for 0 < α < h, the α-set of A( ) p B is ( (A( ) p B) α = {(x, y) R 2 x x + x 2 a (h α) + a 2 (h 2 α) ( + y y + y 2 b (h α) + b 2 (h 2 α) =

A generalization of a generalized triangular fuzzy sets 44 If y = 0, the 0-set of A( ) p B is { ( lim = (x, 0) R 2 x x + x 2 = α 0 +(A( )B)α a h + a 2 h 2 Thus the 0-cut of µ A( )B (x, 0) is [x x 2 (a h + a 2 h 2 ), x x 2 + (a h + a 2 h 2 )] Applying Theorem 25, we have x x 2 (a h + a 2 h 2 ) = m n 2 x x 2 + (a h + a 2 h 2 ) = m 2 n If y = 0, the h -set of A( ) p B is { lim (A( )B) α = α h (x, 0) R 2 ( x x + x 2 a 2 (h 2 h ) = Thus the h -cut of µ A( )B (x, 0) is [x x 2 a 2 (h 2 h ), x x 2 + a 2 (h 2 h )] Applying Theorem 25, we have x x 2 a 2 (h 2 h ) = 2 (m + m 2 ) k 2 x x 2 + a 2 (h 2 h ) = 2 (m + m 2 ) k Since A( )B is a generalized trapezoidal fuzzy set, we have µ A( )B (x, 0) = µ A( )B(x) (3) If y = y 2 = 0, we have (A( ) p B) α = {(x α (t), y α (t)) 0 t 2π, where If y α (t) = 0, t = π Thus x α (t) = x x 2 + (x a 2 (h 2 α) + x 2 a (h α)) cos t + a a 2 (h α)(h 2 α) cos 2 t, 0 < α < h, y α (t) = b b 2 (h α)(h 2 α) sin 2 t, 0 < α < h x 0 (π) = x x 2 (x a 2 h 2 + x 2 a h ) + a a 2 h h 2 = (x a h )(x 2 a 2 h 2 ) x 0 (0) = x 0 (2π) = x x 2 + (x a 2 h 2 + x 2 a h ) + a a 2 h h 2 = (x + a h )(x 2 + a 2 h 2 ) Thus the 0-cut of µ A( )B (x, 0) is [(x a h )(x 2 a 2 h 2 ), (x +a h )(x 2 +a 2 h 2 )] Applying Theorem 25, we have (x a h )(x 2 a 2 h 2 ) = m n (x + a h )(x 2 + a 2 h 2 ) = m 2 n 2 Similarly, we have x h (π) = x x 2 x a 2 (h 2 h ) x h (0) = x h (2π) = x x 2 + x a 2 (h 2 h ) Thus the h -cut of µ A( )B (x, 0) is [x x 2 x a 2 (h 2 h ), x x 2 + x a 2 (h 2 h )] Applying Theorem 25, we have x x 2 x a 2 (h 2 h ) = 2 (m + m 2 )k x x 2 + x a 2 (h 2 h ) = 2 (m + m 2 )k 2 Thus µ A( )B (x, 0) = µ A( )B(x) (4) If y = y 2 = 0, we have (A(/) p B) α = {(x α (t), y α (t)) 0 t 2π, where

442 Chang Il Kim Yong Sik Yun x α (t) = x + a (h α) cos t x 2 a 2 (h 2 α) cos t, y α(t) = b (h α) sin t b 2 (h 2 α) sin t, 0 < α < h We cannot find the value of t which satisfies y α (t) = 0 But if α = h, for all t 0, π, 2π, we have Thus y h (t) = 0 x h (t) = x x 2 a 2 (h 2 h ) cos t lim x h t π (t) = x x 2 + a 2 (h 2 h ) lim x h t 2π (t) = x x 2 a 2 (h 2 h ) Therefore the h -cut of µ A(/)B (x, 0) is [ x x 2 + a 2 (h 2 h ), x x 2 a 2 (h 2 h ) ] Applying Theorem 25, we have x x 2 + a 2 (h 2 h ) = m + m 2 2k 2 Similarly, we have x x 2 a 2 (h 2 h ) = m + m 2 2k x 0 (π) = lim t π x 0(t) = x a h x 2 + a 2 h 2 x 0 (0) = x 0 (2π) = lim t 2π x 0(t) = x + a h x 2 a 2 h 2 Thus the 0-cut of µ A(/)B (x, 0) is [x a h, x + a h ] Applying Theorem x 2 + a 2 h 2 x 2 a 2 h 2 25, we have Thus µ A(/)B (x, 0) = µ A(/)B(x) x a h x 2 + a 2 h 2 = m n 2 x + a h x 2 a 2 h 2 = m 2 n References [] C Kang YS Yun, An extension of Zadeh s max-min composition operator, International Journal of Mathematical Analysis, 9 (205), no 4, 2029-2035 https://doiorg/02988/ijma2055554

A generalization of a generalized triangular fuzzy sets 443 [2] C Kim YS Yun, Zadeh s extension principle for 2-dimensional triangular fuzzy numbers, Journal of Korean Institute of Intelligent Systems, 25 (205), no 2, 97-202 https://doiorg/0539/jkiis20525297 [3] C Kim YS Yun, Parametric operations for generalized 2-dimensional triangular fuzzy sets, International Journal of Mathematical Analysis, (207), no 4, 89-97 https://doiorg/02988/ijma2076229 [4] YS Yun, SU Ryu JW Park, The generalized triangular fuzzy sets, Journal of the Chungcheong Mathematical Society, 22 (2009), no 2, 6-70 [5] HJ Zimmermann, Fuzzy Set Theory - Its Applications, Kluwer- Nijhoff Publishing, Boston-Dordrecht-Lancaster, 985 https://doiorg/0007/978-94-05-753- Received: April, 207; Published: May 3, 207