Preinvexity and Phi1-convexity of fuzzy mappings through a linear ordering

Keywords--Fuzzy numbers, Convexity, Preinvexity, Generalized convexity, Fuzzy mappings. 1. I N T R O D U C T I O N Let R n deno te t he n -d imens iona l Euc l idean space. T h e suppor t , supp (p), of a fuzzy set # : R n ~ I = [0, 1] is def ined as s u p p ( # ) = { x e R n ] # ( x ) > 0} . A fuzzy set # : R n ~ I is cal led fuzzy convex if i t is quas i concave in c o m m o n sense on its suppor t . For a • [0, 1], t h e a l eve l set of a fuzzy set # : R n --* I is def ined as ( {z e Rn },(x) > a}, i f 0 < a < l ; [#la = cl (supp (#)) , if a = 0, where cl ( supp (#)) deno tes t h e c losure of supp (#). A fuzzy set # : R n --* I is said to be no rma l if [.]1 ~ 0. Supported by the National Science Council of the Republic of China under contract NSC 90-2218-E-212-013. This work was carried out while the first author was visiting the Department of Industrial and Manufacturing Systems Engineering, Kansas State University. *Author to whom all correspondence should be addressed. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2005.10.010 Typeset by A~-TF_~ 406 Y.-R. SYAU AND E. S. LEE A fuzzy number we treat in this study is a fuzzy set # : R 1 ~ I which is normal, fuzzy convex. upper semicontinuous and with bounded support. It is clear that each r C R t can be considered as a fuzzy number. Let 5 r denote the set of all fuzzy numbers. A mapping from any nonempty set into .%" will be called a fuzzy mapping. Thus, any real-valued function can be considered as a fuzzy mapping. The concept of convexity, preinvexity, and (I)l-convexity for fuzzy mappings has been considered by many authors in fuzzy optimization. For example, in [1-5], the convexity and quasiconvexity of fuzzy mappings based on the "fuzzy-max" order on 5 r , were investigated. In [6,7], the preinvexity and prequasiinvexity of fuzzy mappings defined through the "fuzzy-max" order on ~" were introduced and studied. In an earlier paper [8], we proposed the (I)vconvexity and ~l-quasiconvexity of fuzzy mappings based on the "fuzzy-max" order on .%-. However, the "fuzzy-max" order is a partial ordering on 5 r. In [9,10], the convexity and quasiconvexity of fuzzy mappings based on a linear ordering proposed by Goetschel and Voxman [11] were investigated. In [9], we defined a ranking value function on 9 r based on the linear ordering proposed in [11]. We extended the local-global minimum property of real-valued convex functions to convex fuzzy mappings, and the Weirstrass theorem from real-valued functions to fuzzy mappings. Motivated both by earlier research works and by the importance of the concepts of convexity and generalized convexity, we introduce the preinvexity, prequasiinvexity, (I)l-convexity, and O1quasiconvexity of fuzzy mappings based on the ranking value function proposed in our earlier work [9]. Characterizations for these fuzzy mappings are given. The local-global minimum property of real-valued preinvex (resp., ~l-convex) functions is extended to preinvex (resp., (biconvex) fuzzy mappings. In addition, it is also proved that every strict local minimizer of a prequasiinvex fuzzy mapping is also a strict global minimizer, and that every strict local minimizer of a (I)l-quasiconvex fuzzy mapping is a strict global minimizer. 2. P R E L I M I N A R I E S In this section, for convenience, several definitions and results without proof from Diamond and Kloeden [12], Goetschel and Voxman [11], Pini and Singhand [13], Syau [7], and Syau and Lee I9] are summarized below. It is known [11] that a fuzzy set # : R 1 ~ I is a fuzzy number if and only if (i) [#]a is a closed and bounded interval for each a e [0, 1], and (ii) [#] 1 ~ 0. Thus, we can identify a fuzzy number # with the parameterized triples, { (a(a) ,b(a) ,a) I O < a < 1}, where a((x) and b(a) denote the leftand right-hand endpoints of [/~]~, respectively. For fuzzy numbers #, v E ~represented parametrically by { (a(a) ,b (a) ,a ) [ 0 <_ a <_ 1} and {(c(a), d(a), a) [ 0 < a < 1}, respectively, and each real number r, we define the addition p + u and scalar multiplication r# as follows, # + p = { ( a ( a ) + c ( a ) , b ( a ) + d ( a ) , a ) 1 0 < a < l } , r p = { ( r a ( a ) , r b ( a ) , a ) ] 0 < a < 1}. (2.1) (2.2) It is known that the addition and nonnegative scalar multiplication on 9 v defined by (2.1) and (2.2) are equivalent to those derived from the usual extension principle, and that ~is closed under the addition and nonnegative scalar multiplication. It should be noted that for each # E ~-, r# is not a fuzzy number for r < 0. However, the family of parametric representations of 9 r and the parametric representations of their negative scalar multiplications form subsets of the vector space, 1) = {{ (a (a) ,b (a) ,a ) I 0 < a < 1} l a : I ~ R 1 and b: I --* R 1 are bounded functions}, Preinvexity and ¢l-Convexity 407 with addit ion and scalar multiplication defined by (2.1) and (2.2). Define a metric D : 12 x 12 [0, oc) by D ( { ( a ( a ) , b ( a ) , a ) [ O < a < 1 } , { ( c ( a ) , d ( a ) , a ) 1 0 < a < 1}) = sup {max {[a (a) c (a)] , Ib(a) d (a)[} l O _< a _< 1}, for { ( a ( a ) , b ( a ) , a ) [ 0 <_ a <_ 1}, { ( c ( a ) , d ( a ) , a ) [ 0 <_ a < 1} 6 12. Then, the vector space Y together with the metric D form a topological vector space. Let 1) = {{(a ( a ) , b ( a ) , a ) [0 _< a <_ 1} 6 121 a and b are Lebesgue integrable}. Goetschel and Voxman [11] defined a ordering on l) as follows. DEFINITION 2.1. (See [11, Definition 2.5].) Suppose t h a t . = { (a (a ) , b (a ) , a ) [ 0 <_ a <_ 1} and . = { ( c ( ~ ) , d ( a ) , ~ ) Io < ~ < 1} are members o f g . Then, ~ Vrecedes ~ (~ ~_ ~) i f ~01 ~0 l a [a (a) + b (a)] da < a [c (a) + d (a)] dct. (2.3) REMARK 2.1. (See [11].) The ordering _ is reflexive and transitive; moreover, any two elements of 9 r are comparable under the ordering _ , i.e., _ is a linear ordering for ~-. DEFINITION 2.2. (See [9].) Let r : J= ~ R 1 be defined by /o' r ( . ) = a [a (a) + b (ct)] da, (2.4) for each. = {(a(a),b(a),a) [0 < a < i} 6.7". LEMMA 2.1. (See [9].) For #, w e jr, and k > O, ( i ) , (~ + ~) = ~ ( . ) + ~-(~,); (2) ~ ( k . ) = k ~ ( . ) . COROLLARY 2.1. For , , v 6 ~ , and kl , k2 > O, T ( k l # + k 2 v ) = k l r ( # ) + k 2 r ( v ) . DEFINITION 2.3. (See [9].) For , , v e J:, we say that # -< v i f , _ u and r (#) # T (V). It is often convenient to write v > , (resp., u >, ) in place o f , ~ v (resp., # -< u). LEMMA 2.2. (See [9].) For #, ~ e ~ , . _ u . = . r ( . ) < ~ (~) , . ~ ~ . = . r ( . ) < ~ (~) . (2.5) (2.6) LEMMA 2.3. (See [9].) For p, v 6 J:, (1) i f # -~ v, t h e n . -.( A. + (1 A)u -~ v for A 6 (0, 1); (2) i f . -'~ u and u ~_ ,, then z(#) = T(V). We have seen that 5 r is closed under addition and nonnegative scalar multiplication. It follows that ~is a convex subset of 12. As mentioned earlier, a mapping from any nonempty set into is a fuzzy mapping. By using the notions of -4 and ~< (resp., the convexity of •), we define the monotonicity (resp., convexity) for a fuzzy mapping 9 : -~ --* ~as follows. 408 Y.-R. SYAU AND E. S. LEE DEFINITION 2.4. (See [9].) A fuzzy mapping g : j r ~ 3: is said to be (1) nondecreasing i f for #, u E jr, (i) # -K v ==> g(#) ~ g(z/); and (ii) T(#) = T(~) ~ 9(#) -= g(v); (2) convex i f for #, ~ E iT and A E (0, 1), r (g (At + (1 A) ~)) _< AT (g (~)) + (1 A) T (g (~)) ; (3) quasiconvex i f for #, u E iT and A E (0, 1), ~" (g (At + (1 A) u)) _< max {T (g (#)) , T (g (V))}. LEMMA 2.4. (See [9, Corollary 4.1].) A fuzzy mapping g : j r --* iT is convex i f and only i f 9 ( A t + (1 A) ~) ~_ Ag (~) + (1 A) g (~), for p, u E iT and A E (0, 1). In what follows, let C be a nonempty convex subset of R n. For any x • R n and 6 > 0, let B~ (x) = {y E R n I ltY xll < 5}, where [I " [I being the 2-norm of R n. DEFINITION 2.5. (See [9].) A fuzzy mapping f : C ~ iT is said to be (1) convex i f for x, y E C and A E (0,1), T ( f (AX + (1 -A) y)) _< AT( f (x)) + (1 A) T ( f (y)) ; (2) quasiconvex i f for x, y E C and A E (0, 1), T ( f (Ax + (1 A) y)) _< max {~( f (x)) ,T ( f (y))}. DEFINITION 2.6. Let S be a nonempty subset of R n. For a fuzzy mapping f : S --~ iT, (1) an element 5: E S is cafled a local minimizer o f f : S ~ iT i f there exists a fi > 0 such that f (5:) -< f (x ) , for all x E S A B~ (~) ; (2) an element ~ E S is called a strict local minimizer of f : S ~ iT i f there exists a 6 > 0 such tha t f ( ~ ) ~ f ( x ) , for a l l x ~ , a n d x E S N B ~ ( ~ ) ; (3) an element x . E S is called a global minimizer of f : S --* jr i f f (x.) _-_g f (x) , for all x E S. Recall [7] that , by definition, a set K C_ R n is said to be an invex set with respect to (w.r.t.) a mapping 7/: K x K -~ R n if x, y E K implies tha t y + A~ (x, y) E K, for A E [0, 1]. In what follows, let S be a nonempty subset of R ~, and let (bl : S x S x [0, 1] ~ R ~ with • l (x ,y ,O) = y, ¢ l ( x , z , A ) = x, for all x, y E S and A E [0,1]. Recall [13] tha t , by definition, a set D C_ S is a (I)l-convex set if ~ l ( x , y , A ) • D for all x, y • D, A • [0, 1]. Preinvexity and ¢l-Convexity 409 3. P R E I N V E X A N D P R E Q U A S I I N V E X F U Z Z Y M A P P I N G S In this section, we introduce the preinvexity and prequasiinvexity of fuzzy mappings based on the ranking value function ~ : 9 v --* R 1 introduced in the preceding section. In what follows, let 77 : K × K --* R n, and let K c_ R n be a nonempty invex set w.r.t, to 77. DEFINITION 3.1. A fuzzy mapping f : K ~ 2: is said to be (1) preinvex i f for x, y E K and A E [0, 1], T ( f ( y + A T l ( x , y ) ) ) ~_ AT ( f (x)) + (1 -A)~" ( f (y)) ; (2) preincave if for x, y E K and X E [0, 1], (S (y + A~ (x, y))) > A~ (f (z)) + (1 A) ~ (f (y)); (3) prequasiinvex i f for x, y E K and A E [0, 1], T ( f (y qAz] (x, y))) _< max {~( f (x)) ,T ( f (y))} ; (4) prequasiincave i f for x, y E K and A E [0, 1], T ( f (y + A~(x ,y) ) ) _> m i n { T ( f ( X ) ) , T ( f (y))}. From Definition 3.1, we obtain the following. LEMMA 3.1. I f f : K --* J: is a preinvex (resp., preincave) fuzzy mapping, then it is also prequasiinvex (resp., prequasiincave). THEOREM 3.1. Let f : K --* j r be a preinvex fuzzy mapping. I f g : ~ ---* ~ is nondecreasing and convex, then the fuzzy mapping g o f : K --* J: defined by (g o f ) (x) = g ( f (x)) , for each x E K, is preinvex on K. PROOF. Let x, y E K and A E [0, 1]. Since f : K ~ 9 r is preinvex, we have T ( f (y + A~ (X, y))) < A~( f (X)) + (1 -A) 7 ( f (y)) . It follows from (2.5) tha t f (y + An (x, y)) ~ AS (z) + (1 A) S (y). (i) f ( A x + (1 A)y) -< A f ( x ) + (1 A)f(y): Since g : ~" --* ~" is nondecreasing and convex, it follows t h a t g ( f (y + Ar] (x ,y))) --< g (Af (x) + (1 A) f (y)) _-< Ag ( f (x)) + (1 A) g ( f (y)) . (ii) -r(f(y + An(x , y))) = "r (A f (x ) + (1 A)f (y)): Since g : ~r ~ ~is nondecreasing and convex, it follows tha t g ( f (y + An (x ,y))) = g (Af (x) + (1 A) f (y)) Ag (S (x)) + (1 A) g (f (y)). From the above arguments and (2.5), we have for x, y ~ K and A E [0, 1], T ( g ( f (y + A~?(x,y)))) < A T ( g ( f (x))) + (1 -A ) ~ ' ( g ( f (y) ) ) , which proves tha t g o f : K --* .~" is preinvex on K. 410 Y . -R . SYAU AND E, S, LEE THEOREM 3.2. Let f j : K --* iT, j = 1 , . . . ,l, be preinvex fuzzy mappings. Fork1, k2,. . . ,kt > O, the fuzzy mapping f : K ~ .T defined by l f (x) = ~ k j f j (x ) , j = l for each x E K, (3.1) is a preinvex fuzzy mapping. PROOF. Since f j : K ---* iT is preinvex for each j = 1 , . . . , l, we have for x, y E K and A E (0, 1), T ( f j (y + A??(x,y))) < AT ( f j (X)) + (1 -A) ~( f j (y ) ) , j = 1 , . . . , l . Then, by Lemma 2.1 (2), it follows t ha t for x, y E K and A E (0, 1), T ( k j f j (y + A~ (x, y))) < A~( k j f j (x)) + (1 A) r ( k j f j (y ) ) , j = 1 , . . . , I. From (3.1) and Lemma 2.1, it follows t ha t for x, y E K and A E (0, 1), 0 + x n ( z , v ) ) ) kjfjO+Av(z,v)) l = T ( k j 5 (v + y))) j = l l < [AT (kj& + (1 x) T (k j f j (v))] j----1 = A'r k i f j (x + (1 A) T k j f j (y = A~" ( f (z)) + (1 A) ~" ( f (y ) ) , which proves t ha t f : K --* 5 r is a preinvex fuzzy mapping. THEOREM 3.3. Let f : K ~ S: be preinvex. I f h~ E K is a local minimizer o f f , then 5: is also a global minimizer of f over K . PROOF. The proof is by contradict ion. Let 5: E K be a local minimizer of f : K --* iT and suppose, by contradict ion, t ha t it is not a global minimizer. Then, there exists some point x E K satisfying f ( x ) -< f (~ ) . Then, by (2.6), we have T ( f ( x ) ) < T(f(hc)). Since f : K -~ iT is preinvex, it follows t ha t T(f(5: + A~?(x, ~c))) < AT( f (X)) + (1 A)r(f( :~)) < T( f (£ ) ) , for all A E (0, 1]. Then, by (2.6), we have f ( ~ + A~(x,~)) -~ f ( ~ ) for a rb i t r a ry small posit ive number A, and this contradic t ion proves the result. THEOREM 3.4. Let f : K ~ iT be prequasiinvex, and let ~(x, y) # O, where O being the origin of R n, for all x, y E R n, x # y. I f • E K is a strict local minimizer o f f , then 3c is also a strict global minimizer of f over K . PROOF. Let ~ E K be a s tr ict local nfinimizer of f : K -~ iT. Then, there exists a ~f > 0 such t ha t f(]:) -< f ( x ) , for all z # ~ and z E S N B~(~). Preinvexity and ¢l-Convexity 41.1 The proof is by contradiction. Suppose, by contradiction, tha t ~ is not a strict global minimizer. Then there exists some point x ' 6 K satisfying f ( x ' ) ___ f (~) . Then, by (2.5), we have T( f ( x ' ) ) < ~'(f(~)). Since f : K ~ ~" is prequasiinvex, it follows tha t T (f (~: + An (x, ~:))) _< max {T (f (x)), T (f (:~))} T (f (~)) , for all A • (0, 1]. Then, by (2.5), we have f(& + An(x, k)) _-_g f (2) . But for sufficiently small A > 0, it follows tha t x ~ ~ and x • S n B~ (5). This contradict ion proves the result. In order to include singletons in R n as invex sets, from now on, we will also assume tha t for all x E R n, n (z, x) = O, (3.2) where O being the origin of R n. THEOREM 3.5. Let K be a nonempty invex subset o f R ~, and let f be a fuzzy mapping on K. The following conditions are equivalent. (1) f : K --* .7 = is preinvex; (2) for x, y E K and A E [0, 1], : (y + An (x, y)) ~ AS (~) + (1 A) f (y) ; (3.3) (3) the epigraph, e p i ( f ) = { ( x , p ) i x E K , # E 9 v, f ( x ) _ ~ # } , o f f : K ~ J: is an invex subset o f R n x V w.r.t, the mapping, (3.4) n ' : epi ( f ) x epi ( f ) ---* R n x V, defined by n' ( ( z , . ) , ( y , . ) ) = (7 (x, y) , ~ + ( -1) . ) , (3..5) for (x, #), (y, u) E epi ( f ) with x, y E K and #, u E F . PROOF. (1) ~ (2). Let f : K --* 5 tbe preinvex, and let x, y E K. Then, for A E [0, 1], we have ( I (y + An (z, y))) < A~ ( : (z)) + (1 A) r (S (y)). From Corollary 2.1, it follows tha t "r( f (y + A n ( x , y ) ) ) ~_ 7"(Af (x) + ( 1 A) f (y ) ) , for A • [0, 1]. Then, by (2.5), we have I (y + An (x, y)) ~ AS (x) + (1 A) ] (y), for A E [0, 1], which proves (3.3). (2) ~ (3). Assume tha t f : K ~ ~" satisfies (3.3). If epi ( f ) is the emp ty set, then it is obvious an invex set. If epi ( f ) is a singleton, then it can be easily checked from (3.2) tha t the singleton is an invex set w.r.t, n'. Let (x ,#) , (y, u) E epi ( f ) , where x, y E K , and #, u E 5 r. Then, from (3.4), we have f ( x ) ~ # and f ( y ) -~ u. I t follows from (2.5) tha t ~( f (x)) ~ r (~) and T ( f (y)) ~ T (V). 412 Y.-R. SYAU AND E. S. LEE From (3.3), (2.5), and Corollary 2.1, it follows tha t ( f (v + ~n (z, v))) -< r (h i (z) + (1 ~) f (v)) = ~ . ( f (z)) + (1 ~ ) . ( f (v)) <_ A~(#) + (1 A) T (u) = 7(A/~ + ( 1 A) v ) , for each A • [0, 1]. Then, by (2.5), we have for all A • (0, 1), f (y + An (x, y)) _~ ~ + (1 ~) v, which implies tha t for each A • [0, 1], (y + A~ (z, y) , ,k# + (1 A) v) = (y,v) + A ( ~ ( x , y ) , p + ( 1 ) v) • epi ( f ) . This proves tha t epi ( f ) is an invex subset of R n x 1; w.r.t, the mapping ~7' defined by (3.5). (3) ==~ (1) Assume tha t epi ( f ) is an invex set w.r.t, the mapping ~?' defined by (3.5). Let x. y • K. Then, from the reflexivity of -< and (3.4), we have (x, f ( x ) ) , (y, f ( y ) ) • epi ( f ) . By the invexity of epi ( f ) , it follow tha t for all A E [0, 1], (y, f (y)) + A (77 (x, y) , f (z) + ( 1 ) f (y)) = (y + AT/ (x, y) , Af (x) + (1 A) f (y)) • epi ( f ) , which implies tha t for all A E [0, 1], f (y + An (z, y)) _~ ~ f (z) + (1 ~) f (y). Then, by (2.5) and Corollary 2.1, we have for all A E [0, 1], r ( f (v + ~ (z, v))) < r (~f (x) + (1 ~) f (v)) = AT ( f (x)) + (1 -A) T ( f ( y ) ) , which completes the proof. THEOREM 3.6. A fuzzy mapping f : K --* ~ is prequasiinvex w.r.t. ~? i f and only i f for each p E .~, the lower #-level set, L u ( f ) = {z • K [ f ( z ) _ p} , (3.6) o f f : K --* 5 c is an invex subset o f R n w.r.t. 77. PROOF. Assume tha t f : K --* ~ is prequasiinvex, and let # • ~'. If L v ( f ) is the empty set or a singleton, then it is obvious an invex set. Let x, y • L v ( f ) . From (3.6), we have f ( x ) ~_ # and f ( y ) -< #. Then, by (2.5), we have T ( f (X)) < T (/~) and , ( f (y)) < T (#) . Since f : K --* ~" is prequasiinvex, from (2.5), we have r ( f (y + A~? (x, y))) < max {r ( f (x)) ,T ( f (y))} _< ~ (p) , for all A • [0, 1]. It follows from (2.5) tha t for each A • [0, 1], f (v + ~,n(z,v)) ~_ #, which implies tha t y + X~(x, y) E L~,(f) for each X • [0,1]. Hence, Lv(. f ) is an invex set w.r.t. 77. Preinvexity and ~l-Convexity 413 Conversely, assume tha t L~,(f) is an invex subset of R n w.r.t, rl for each # E 5 c. Let x, y E K. Without loss of generality, we may assume that f ( x ) "< f ( y ) . Let # = f ( y ) . Since _-< is reflexive and transitive, we have f (x) ~_ # and f (y) _~ #, which implies tha t x, y E L~,(f). Then, by the invexity of L~,(f), we have y + A~?(x, y) E L , ( f ) for each A E [0, 1], which implies tha t y (y + ~ (x, u)) _~ ~, for all ~ • [0, 1]. Since f ( x ) -< f ( y ) and # --f ( y ) , from (2.5), we have, for all A • [0, 1], 7(I (v + ~v (x, v))) < 7(~) = max {7(f (X)),T (f (y))}, which complete the proof. THEOREM 3.7. Let f : K ~ ~" be a prequasiinvex fuzzy mapping w.r.t. ~, and let x , • K be a global minimizer o f f over K . Then, the set, = {z • K 17( : (~)) = 7( : ( z . ) ) } , is an invex set w.r.t. ~. PROOF. If f2 is a singleton, then it is obvious an invex set. Assume tha t x, y • f2. Then, we have 7( f (x)) = 7( f (x.)) and 7( f (y)) = -r ( f (x.)) . Since f : K --* 9 v is a prequasiinvex fuzzy mapping w.r.t. ~, we have for all A • [0, 1], 7(f (Y + A~ (x, Y))) -< AT(f (x)) + (I :~) ~" (f (Y)) = T (: (z,)). Since x . • K is a global minimizer of f : K --* 9 v, it follows from Lemma 2.3 (2) tha t 7(f (v + .~,7 (~, v))) = ~(f (x.)), for all A • [0, 1], which implies tha t y + A~?(x,y) • f~ for all A • [0, 1]. Hence, f2 is an invex set w.r.t. 77. This complete the proof. From Theorem 3.7 and Lemma 3.1, we obtain the following. THEOREM 3.8. Let f : K --* Jr be a preinvex fuzzy mapping w.r.t. ~, and let x . • K be a global minimizer of f over K . Then, the set, f~ = {x • K I 7( f (z)) = 7" ( f ( z , ) ) } , is an invex set w.r.t. ~. 4. ( I ) I C O N V E X A N D ¢ I ) I Q U A S I C O N V E X F U Z Z Y M A P P I N G S In this section, we introduce the (I)l-convexity and (I)l-quasiconvexity of fuzzy mappings based on the ranking value function T : : " ~ R 1. In what follows, let D C S be a nonempty ~l-convex set. 414 Y.-R. SYAU AND E. S. LEE DEFINITION 4.1. A fuzzy mapping f : D --* jc is said to be (1) e l convex if for x, y • D and ~, • [0,1], 7( f (~1 (x, y, A))) < AT ( f (x)) + (1 -A) z ( f (y)) ; (2) el-COnCave i f for x, y • D and A • [0, 1], ~" ( f (¢1 (x, y, A))) _> A-r ( f (x)) + (1 A) T ( f (y)) ; (3) ¢l-quasiconvex if for x, y • D and A • [0,1], r ( f (¢1 (x, y, )~))) <_ max {z ( f (x)) , r ( f (y))} ; (4) q?l-quasiconcave i f for x, y 6 D and A 6 [0, 1], z ( f (¢ i (X, y, A))) _> min {~( f (x)), r ( f (y))}. From Definition 4.1, we obtain the following. LEMMA 4.1. I f f : D --* j r is a e l convex (resp., el-concave) fuzzy mapping, then it is also ¢l-quasiconvex (resp., ¢l-quasiconcave). THEOREM 4.1. Let f : D ~ Jr be a e l convex fuzzy mapping. I f g : J: --~ J: is nondecreasing and convex, then the fuzzy mapping g o f : D ~ :7: defined by (g o f ) (x) = g ( f (x)), for each x • D, is ~ l-convex on D. PROOF. Let x, y • D and A • [0, 1]. Since f : D -* ~" is el-convex, we have T ( f ((I) 1 (Z, y, ,~))) <: ,~T ( f (X)) + (1 -/~) T ( f (y)). It follows from (2.5) that f (¢1 (x,v,A)) _--< Af (x) + (1 A) f (V)(i) f ( ~ l ( x , y , A ) ) -< A f ( x ) + (1 A)f(y): Since g j r __~ Dr is nondecreasing and convex, it follows tha t g ( f (¢ , (x, y, A))) ~ g (Af (x) + (1 A) f (y)) ;~g ( f (~)) + (i ;~) g ( f (v)). (ii) "r ( f (¢ l (x , y, A))) = "r(Af(x) + (1 A)f(y)): Since g : ~~ :" is nondecreasing and convex. it follows tha t g ( f (¢1 (x, y, A))) = g (Af (x) + (1 A) f (y)) __ Ag ( f (x)) + (1 A) g ( f (y)). From the above arguments and (2.5), we have for x, y 6 D and A 6 [0, 1], T (g ( f (¢1 (x, y, A)))) _< AT (g ( f (x))) + (1 A) T (g ( f (y))), which proves that g o f : D ~ 9 r is ¢l-convex on D. Preinvexity and ¢l-Convexity Let f j : D --* J=, j = 1 , . . . , l, be ¢ l -convex fuzzy mappings. 415 For kl, k'2 . . . . . and defined by ¢ 1 ' : epi ( f ) x epi ( f ) x [0,1] ~ R n x V, ¢1' ((x, # ) , (y, v ) , A) = (¢1 (x, y, A), A# + (1 A) v) , (4.2) for A E [0,1] and (x ,#) , (y,u) E e p i ( f ) with x, y E D and #, v • 2 r. It can be easily checked tha t ¢1' : epi ( f ) x epi ( f ) x [0, 1] ~ R n x )2 satisfies ¢1' ( (x ,# ) , ( y , v ) , 0 ) --(¢1 (x,y, 0 ) ,v ) = (y ,v) (4.3) ¢1' ((X, # ) , (x, # ) , A) = (¢1 (x, x, A), A# + (1 A) #) = (x, # ) , (4.4) for all (x, #), (y, v) E epi ( f ) and A E [0, 1]. (1) ==~ (2) Let f : D -~ ~" be ¢l-convex, and let x, y • D. Then for A • [0, 1], we have r ( f (¢1 (X, y, X))) <_ A~ ( f (X)) + (1 -X) ~ ( f (y)) . From Corollary 2.1, it follows tha t ( f (¢1 (z, v, ~))) < ~ (hi (z) + (1 ~) f (v)), for A E [0, 1]. Then by (2.5), we have for A • [0, 1], f (¢1 (X, y, )~)) "~ ~ f (X) -~(1 A) f (y) , which proves (4.1). (2) ~ (3) Assume tha t f : D ~ 9 r satisfies (4.1). If epi ( f ) is the empty set or a singleton, then, from (4.3) and (4.4), it is obvious a ¢ l ' -convex subset of R n x V. Let (x ,#) , (y, v) E epi ( f ) , where x, y E D, and #, ~, E ~'. It follows tha t f ( x ) ~_ p and f ( y ) ~_ ~. Then, by (2.5), we have ~( f (x)) < T (/~) and r ( f (y)) < ~(g). PROOF. THEOREM 4.2. kz > 0, the fuzzy mapping f : D --* ~ defined by l f (x) = ~ k j f j (x ) , for each x ~ D, j=l is a e l convex fuzzy mapping. PROOF. The idea of the proof is similar to tha t of Theorem 3.2. THEOREM 4.3. Let D be a nonempty e l convex subset o f R n, and let f be a fuzzy mapping" on D. The following conditions are equivalent. (1) f : D --* 3 ~ is e l -convex; (2) for x, y E D and A E [0,1], f ((I)l (x,y,A)) ~ Af (x ) + (1 A) f (y) ; (4.1) (3) the epigraph e p i ( f ) = ( ( x , p ) [ x E D , # E 5 r , f ( x ) ~ # } o f f : D -* Y is a ¢ l ' -convex subset of R n × )2 w.r.t, the mapping, 416 Y.-R. SYAU AND E. S. LEE From (4.1), (2.5) and Corollary 2.1, it follows tha t "r (f (~i (x,y,A))) < "r (Af (x) + (I A) f (y)) = AT (f (x)) + (I -A) T (f (y)) _< (.) + (1 (.) = r(A#+ (i A) u), for each A E [0, 1]. Then, by (2.5), we have for all A E [0, 1], f(~I(x,Y,A))_AP+(I-A)",