European Journal of Operational Research 182 (2007) 965–970

www.elsevier.com/locate/ejor

Short Communication

Pseudolinear fuzzy mappings q

S.K. Mishra a,*, S.Y. Wang b, K.K. Lai c

a Department of Mathematics, Statistics and Computer Science College of Basic Sciences and Humanities,

G.B. Pant University of Agriculture and Technology, Pantnagar, Indiab Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Humanities, Beijing, China

c Department of Management Sciences, City University of Hong Kong, 83-Tat Chee Avenue, Kowloon, Hong Kong

Received 23 July 2004; accepted 19 September 2006Available online 13 November 2006

Abstract

Two classes of fuzzy mappings, called pseudolinear and g-pseudolinear fuzzy mappings are introduced by relaxing thedefinitions of pseudo-convex and pseudo-invex fuzzy mappings. First, some characterizations of pseudolinear and g-pseudolinear fuzzy mappings are obtained. Then, characterizations of the solution sets of pseudolinear and g-pseudolinearfuzzy programs are derived.� 2006 Elsevier B.V. All rights reserved.

Keywords: Fuzzy numbers; Convex fuzzy mappings; Pseudolinear fuzzy mappings

1. Introduction

Several convexity concepts [1–18] has been considered by many authors in optimization. In [11,19,20,22,23,25] different types of convexity and generalized convexity of fuzzy mappings are defined and theirproperties are studied. Equally important is the concept of pseudolinearity and g-pseudolinearity, which wereintroduced and studied in [1,4,7,9,10].

We propose in this paper the concept of pseudolinear fuzzy mappings by relaxing the definition of pseudo-convex fuzzy mappings. The concept of g-pseudolinear fuzzy mappings is also introduced by relaxing the def-inition of pseudo-invex fuzzy mappings. By means of the basic properties of pseudolinear fuzzy mappings, thesolution set of a pseudolinear fuzzy program is characterized. Then, characterizations of the solution set of ang-pseudolinear program are also derived.

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2006.09.033

q This research is supported by the Council of Scientific and Industrial Research, New Delhi, through Grant No. 25 (0132)/04/EMR-II,National Natural Research Foundation of China and the Research Grants Council of Hong Kong.

* Corresponding author.E-mail address: shashikmishra@rediffmail.com (S.K. Mishra).

966 S.K. Mishra et al. / European Journal of Operational Research 182 (2007) 965–970

2. Preliminaries

Let Rn be n-dimensional Euclidean space. Following Syau [25] we recall: A fuzzy number is a fuzzy setu: R1! [0,1] which is normal, fuzzy convex, upper semicontinuous and with bounded support. Let I0 denotethe family of fuzzy numbers. Since each r 2 R1 can be considered as a fuzzy number ~r defined by

~rðtÞ ¼1 if t ¼ r;

0 if t 6¼ r

�

R1 can be embedded in I0. As known [25] the a-level set of a fuzzy number u 2 I0 is a closed and bounded

interval:

½u�ðaÞ; uþðaÞ� ¼ ½u�a ¼fx 2 R1 : uðaÞP ag if 0 < a < 1;

clðsupp uÞ if a ¼ 0:

(

It is easily verified that a fuzzy set u: R1! [0, 1] is a fuzzy number if and only if (i) [u]a is a closed and boundedinterval for each a 2 [0,1] and (ii) [u]1 5 /. Thus, we can identify a fuzzy number u with the parametrized tri-ples {(u�(a), u+(a),a): 0 6 a 6 1}, where u�(a) and u+(a) denote the left- and right-hand endpoints of [u]a,respectively. For fuzzy numbers u; v 2 I0 represented by {(u�(a), u+(a),a): 0 6 a 6 1} and {(v�(a), v+(a),a):0 6 a 6 1}, respectively and each real number r, we define the addition u + v and scalar multiplication ru

as follows:

uþ v ¼ fðu�ðaÞ; uþðaÞ; aÞ : 0 6 a 6 1g þ fðv�ðaÞ; vþðaÞ; aÞ : 0 6 a 6 1g¼ fðu�ðaÞ þ vþðaÞ; uþðaÞ þ vþðaÞ; aÞ : 0 6 a 6 1g;

ru ¼ fðru�ðaÞ; ruþðaÞ; aÞ : 0 6 a 6 1g:

It should be noted that for u 2 I0; ru is not a fuzzy number for r < 0. The family of parametric representationsof members of I0 and the parametric representations of their negative scalar multiplications from subsets ofthe vector space @ = {{(u�(a),u+(a),a): 0 6 a 6 1}}; u�: [0,1]! R1 and u+: [0, 1]! R1 are bounded functions.

We metricize @ by the metric

dðfðu�ðaÞ; uþðaÞ; aÞ : 0 6 a 6 1g; fðv�ðaÞ; vþðaÞ; aÞ : 0 6 a 6 1gÞ¼ supfmaxfju�ðaÞ � v�ðaÞj; juþðaÞ � vþðaÞjg : 0 6 a 6 1g:

Let @� = {{(u�(a),u+(a),a): 0 6 a 6 1}; u� and u+: [0, 1]! [0,1] are bounded functions.It is obvious that I�0 � @

� � @, where I�0 denotes the set of all nonnegative fuzzy numbers of I0. It is easilyverified [24] that @� is a closed convex cone in the topological vector space ð@; dÞ. Suppose that u; v 2 I0 arefuzzy numbers represented by {(u�(a), u+(a),a): 0 6 a 6 1} and {(v�(a),v+(a),a): 0 6 a 6 1}, respectively.Define a partial ordering � in I0 by u � v if and only if u�(a) 6 v�(a) and u+(a) 6 v+(a) for all a 2 [0,1].We say that u � v if u � v and there exists a0 2 [0,1] such that u�(a0) < v�(a0) and u+(a0) < v+(a0). We say thatu = v if and only if u � v and v � u. It is often convenient to write v � u (respectively v u) in place of u � v

(u � v). For u; v 2 I0, it is clear that u � v 2 @� if and only if u � v and that u� v 2 @�nf~0g iff u v. It is also

obvious that addition and nonnegative scalar multiplication preserve the order on I0. A fuzzy numberu: R1! [0,1] is called nonnegative if u(t) = 0 for all t < 0. It can be seen easily that for u 2 I0, u is nonnegativeif and only if u � ~0. A subset S* of I0 is said to be bounded above if there exists a fuzzy number u 2 I0 calledan upper bound of S*, such that v � u for every v 2 S*. Further, a fuzzy number u0 2 I0 is called the leastupper bound (supp in short) for S* if (i) u0 is an upper bound of S* and (ii) u0 � u for every upper boundu of S*. A lower bound and the greatest lower bound (inf in short) are defined similarly.

It is known [21] that every non-negative set S� I0 which is bounded above (respectively bounded below)has a least upper (respectively greatest lower) bound. In particular, sup{u,v} and inf{u,v} exist in I0 for everypair fu; vg � I0: Furthermore, according to Syau [21], we have inf{u,v} � ku + (1 � k)v � sup{u,v} for everyu; v 2 I0 and k 2 [0, 1]. Finally, we give preliminary definitions of differentiable fuzzy mappings of several vari-ables and fuzzy mappings from the standpoint of convex analysis. Let K be a nonempty convex subset of Rn.Let T be a nonempty open and convex subset of Rn.

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Definition 2.1 (Syau [25]). A fuzzy mapping F : T ! I0 is differentiable at x0 2 T if there is a unique lineartransformation k : Rn ! @ such that

limkDxk!0

1

kDxk dðF ðx0 þ DxÞ; F ðx0 þ kDxÞÞ ¼ 0;

where k Æk denotes the usual Euclidean norm in Rn.

Remark 2.1. The linear transformation k is denoted by DðF Þx0and called the differential of F at x0.

Definition 2.2 (Syau [25]). A differentiable fuzzy mapping F : T ! I0 is called pseudo-convex if for allx; y 2 T ;DðF Þy ðx� yÞ 2 @� ) F ðxÞ � F ðyÞ 2 @�

F : T ! I0 is called pseudo-concave if for all x,y 2 T,

�DðF Þy ðx� yÞ 2 @� ) F ðyÞ � F ðxÞ 2 @�:

Definition 2.3. A differentiable fuzzy mapping F : T ! I0 is called pseudolinear if for all x,y 2 T, F and �F arepseudo-convex, i.e.

DðF Þy ðx� yÞ 2 @� ) F ðxÞ � F ðyÞ 2 @� and � DðF Þy ðx� yÞ 2 @� ) F ðyÞ � F ðxÞ 2 @�:

Definition 2.4. A differentiable fuzzy mapping F : T ! I0 is called pseudo-invex with respect to a functiong: T · T! Rn if for all x,y 2 T,

DðF Þy ðgðx; yÞÞ 2 @� ) F ðxÞ � F ðyÞ 2 @�:

F : T ! I0 is called pseudo-incave with respect to a function g: T · T! Rn if for allx; y 2 T ;�DðF Þy ðgðx; yÞÞ 2 @� ) F ðyÞ � F ðxÞ 2 @�.

Definition 2.5. A differentiable fuzzy mapping F : T ! I0 is called g-pseudolinear with respect to a functiong: T · T! Rn if for all x,y 2 T, F and �F are pseudo-invex, that is,

DðF Þy ðgðx; yÞÞ 2 @� ) F ðxÞ � F ðyÞ 2 @� and � DðF Þy ðgðx; yÞÞ 2 @� ) F ðyÞ � F ðxÞ 2 @�:

Recall, a set X Rn is said to be invex with respect to g: X · X! Rn if for x,y 2 X, we have y + kg(x,y) 2 X,0 6 k 6 1. Note that every pseudolinear fuzzy mapping is g-pseudolinear with g(x,y) = x � y, but the converseis not true. For the converse part, consider a fuzzy mapping F : T ! I0 in two variables, by F(x) = x1 + sinx2

for all x = (x1,x2) 2 D, where T ¼ fðx1; x2Þ 2 R� R : x1 > �1; p2< x2 <

p2g and gðx; yÞ ¼ ðx1 � y1;

sin x2�sin y2

cos y2ÞT.

Then, F is g-pseudolinear, but not pseudolinear; as at x ¼ ðp3; 0Þ and y ¼ ð0; p

3Þ, then DðF Þy ðx� yÞ ¼ 0, but

F(x) � F(y).

3. Characterizations of pseudolinear fuzzy mappings

In this section, we provide some characterizations of g-pseudolinear fuzzy mappings and pseudolinear fuzzymappings.

Proposition 3.1. Let F be a differentiable fuzzy mapping defined on an open set T in Rn and K be an invex subset

of T such that g: K · K! Rn satisfies

gðx; y þ tgðx; yÞÞ ¼ ð1� tÞgðx; yÞ and gðy; y þ tgðx; yÞÞ ¼ �tgðx; yÞ for all t 2 ½0; 1�:

Suppose that F is g-pseudolinear on K. Then, for all x,y 2 K, DðF Þy ðgðx; yÞÞ ¼ 0 if and only if F(x) = F(y).

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Proof. Suppose that F is g- pseudolinear on K. Then, for all x,y 2 K, we have

DðF Þy ðgðx; yÞÞ 2 @� ) F ðxÞ � F ðyÞ 2 @� and � DðF Þy ðgðx; yÞÞ 2 @� ) F ðyÞ � F ðxÞ 2 @�:

Combining these two, we obtain DðF Þy ðgðx; yÞÞ ¼ 0) F ðxÞ ¼ F ðyÞ for all x,y 2 K.Now, we prove that F ðxÞ ¼ F ðyÞ ) DðF Þy ðgðx; yÞÞ ¼ 0; 8x; y 2 K. For that, we show that for any x,y 2 K

such that F(x) = F(y) implies that F(y + tg(x,y)) = F(y) for all t 2 (0,1). If F(y + tg(x,y)) F(y), then by thedefinition of pseudo-invexity of F, we have DðF Þz ðgðx; zÞÞ � 0, where z = y + tg(x,y). We show thatgðy; zÞ ¼ �t

1�t gðx; zÞ. From the assumption of proposition, we have

gðy; zÞ ¼ gðy; y þ tgðx; yÞÞ ¼ �tgðx; yÞ ¼ �t1� t

gðx; zÞ:

Therefore, from (1), we obtain DðF Þz�t

1�t gðx; zÞ� �

� 0 and hence DðF Þz ðgðx; zÞÞ 0: By g-pseudolinearity of F, wehave F(x) � F(z). This contradicts the assumption that F(z) F(y) = F(x). Similarly, we can also show thatF(y + tg(x,y)) F(y) leads to a contradiction, using pseudo-invexity of �F. Hence, F(y + tg(x,y)) = F(y)for all t 2 (0,1). Thus, DðF Þy ðgðx; yÞÞ ¼ 0. h

Corollary 3.1. Let F be a differentiable fuzzy mapping defined on an open set T in Rn and K be a convex subset of

T. Suppose that F is pseudolinear on K. Then, for all x,y 2 K, DðF Þy ðx� yÞ ¼ 0 if and only if F(x) = F(y).

Remark 3.1. The above corollary extends Proposition 2.2 of Chew and Choo [4] and above Proposition 3.1extends Proposition 1 of Ansari et al. [1] to the case of fuzzy mappings.

Proposition 3.2. Let F be a differentiable fuzzy mapping defined on an open set T in Rn and K be an invex subset

of T. Then F is g-pseudolinear if and only if there exists a function p defined on K · K such that p(x,y) 0 and

F ðxÞ ¼ F ðyÞ þ pðx; yÞDðF Þy ðgðx; yÞÞ, for all x,y 2 K.

Proof. Let F be an g-pseudolinear function. We have to construct a function p on K · K such that p(x,y) 0and F ðxÞ ¼ F ðyÞ þ pðx; yÞDðF Þy ðgðx; yÞÞ for all x,y 2 K. If DðF Þy ðgðx; yÞÞ ¼ 0, for all x,y 2 K, then we definep(x,y) = 1. In this case we have F(x) = F(y), due to Proposition 3.1. On the other hand if DðF Þy ðgðx; yÞÞ 6¼ ~0,then we define pðx; yÞ ¼ F ðxÞ�F ðyÞ

DðF Þy ðgðx;yÞÞWe have to show that p(x,y) 0. Suppose that F(x) F(y). Then, by

pseudo-invexity of �F, we have DðF Þy ðgðx; yÞÞ 0. Hence, pðx; yÞ ~0.For the converse part, we first show that F is pseudo-invex i. e., for any x; y 2 K;DðF Þy ðgðx; yÞÞ 2 @�, then we

have F ðxÞ � F ðyÞ ¼ pðx; yÞDðF Þy ðgðx; yÞÞ � 0. Thus, F(x) � F(y). Likewise, we can prove that �F is pseudo-invex. Hence, F is g-pseudolinear. h

Corollary 3.2. Let F be a differentiable fuzzy mapping defined on an open set T in Rn and K be a convex subset of

T. Then F is pseudolinear if and only if there exists a function p defined on K · K such that p(x,y) 0 and

F ðxÞ ¼ F ðyÞ þ pðx; yÞDðF Þy ðx� yÞ, for all x,y 2 K.

Proof. Replace g(x,y) = x � y in the proof of Proposition 3.2. h

Remark 3.2. Proposition 3.2 and Corollary 3.2 are extensions of Proposition 2 and Proposition 2.1 of Ansariet al. [1] and Chew and Choo [4], respectively, to the case of fuzzy mappings.

Proposition 3.3. Let F: T! Rn be an g-pseudolinear fuzzy mapping defined on an open set T of Rn and G: R! R

be differentiable fuzzy mapping with DðF ÞðtÞ ~0 or DðF ÞðtÞ � ~0 for all t 2 R. Then, the composition mapping F � G

is also g-pseudolinear fuzzy mapping.

Proof. Let H(x) = G(F(x)) for all T. It suffices to prove the result for D(F)(t) 0, since the negative of an g-pseudolinear function is g-pseudolinear. We have, DðHÞy ðgðx; yÞÞ ¼ DðGÞðF ðxÞÞDðF Þy ðgðx; yÞÞ. Then,DðHÞy ðgðx; yÞÞ � ~0 ð� ~0Þ implies DðHÞy ðgðx; yÞÞ � ~0 ð� ~0Þ since G is strictly increasing. This yields F(x) � F(y)

S.K. Mishra et al. / European Journal of Operational Research 182 (2007) 965–970 969

(F(x) � F(y)), due to g-pseudolinearity of F. Thus, H(x) � H(y) (H(x) � H(y)) since G is strictly increasing.Hence, H is g-pseudolinear. h

Corollary 3.3. Let F: T! Rn be a pseudolinear fuzzy mapping defined on an open set T of Rn and G: R! R be

differentiable fuzzy mapping with DðF ÞðtÞ ~0 or DðF ÞðtÞ � ~0 for all t 2 R. Then, the composition mapping F � G isalso pseudolinear fuzzy mapping.

4. Characterization of solution sets

Let F be a fuzzy mapping defined on a nonempty open subset T of Rn, and K is an invex set of T such thatinf{F(x): x 2 K} exists in I0. Let l = inf{F(x): x 2 K}. Syau [25] has established that: C = {x 2 K: F(x) = l} isan invex set if F is preinvex fuzzy mapping. We establish the following result:

Theorem 4.1. Let F : T ! I0 be differentiable fuzzy mapping on an open set T of Rn, and F be g-pseudolinear on

an invex subset K of T where g satisfies g(x,y) + g(y,x) = 0, g(y,y + tg(x,y)) = � tg(x, y) and g(x,y + tg(x,y)) = (1 � t)g(x,y) for all x,y 2 K and for all t 2 [0,1]. Let �x 2 C. Then, C ¼ eC ¼ bC, whereeC ¼ fx 2 K : DðF Þx ðgðx;~xÞÞ ¼ ~0g and bC ¼ fx 2 K : DðF Þx ðgðx;�xÞÞ ¼ ~0g:

Proof. The point x 2 C if and only if F ðxÞ ¼ F ð�xÞ. By Proposition 3.1, we have F ðxÞ ¼ F ð�xÞ if and only ifDðF Þx ðgðx; yÞÞ ¼ ~0. Also F ð�xÞ ¼ F ðxÞ if and only if DðF Þx ðgðx;�xÞÞ ¼ ~0. The later is equivalent toDðF Þx ðgð�x; xÞÞ ¼ ~0, since gð~x; xÞ ¼ �gðx;~xÞ. h

Corollary 4.1. Let F and g be the same as in Theorem 4.1. Then C ¼ eC1 ¼ bC1, whereeC1 ¼ fx 2 K : DðF Þx ðgðx;~xÞÞ � ~0g and bC1 ¼ fx 2 K : DðF Þx ðgðx;�xÞÞ � ~0g.

Proof. It is clear from Theorem 4.1 that C � eC1 We prove that eC1 � C: Assume that x 2 eC1, i.e. x 2 K suchthat DðF Þx ðgðx;�xÞÞ � ~0. In view of Proposition 3.2, there exists a function p defined on K · K such thatpðx;~xÞ 0 and F ðxÞ ¼ F ð�xÞ þ pðx;�xÞDðF Þx ðgðx;�xÞÞ � F ð�xÞ: This implies that x 2 C, and hence eC1 � C. Similarly,we can prove that C ¼ bC1, using the identity gðx;�xÞ ¼ �gð�x; xÞ. h

Corollary 4.2. Let F : T ! I0 be differentiable fuzzy mapping on an open set T of Rn, and F be pseudolinear on a

convex subset K of T. Let �x 2 C. Then, C ¼ eC ¼ bC, where C ¼ sol. set of pseudolinear programme,eC ¼ fx 2 K : DðF Þx ðx� ~xÞ ¼ ~0g and bC ¼ fx 2 K : DðF Þx ðx� �xÞ ¼ ~0g.

Proof. The proof is similar to the proof of Theorem 4.1 with gðx;�xÞ ¼ x�x. h

Corollary 4.3. Let F be as in Corollary 4.2; let �x 2 C. Then C ¼ eC1 ¼ bC1, where eC1 ¼ fx 2 K : DðF Þx ðx� ~xÞ � ~0gand bC1 ¼ fx 2 K : DðF Þx ðx� �xÞ � ~0g.

Theorem 4.2. Let F and g be the same as in Theorem 4.1. If �x 2 C, then C ¼ C� ¼ C�1, whereC� ¼ fx 2 K : DðF Þx ðgðx;�xÞÞ ¼ DðF Þx ðgð�x; xÞÞg and C�1 ¼ fx 2 K : DðF Þx ðgðx;�xÞÞ � DðF Þx ðgð�x; xÞÞg.

Proof

(i) C � C*. Let x 2 C. It follows from Theorem 4.1 that DðF Þx ðgðx;�xÞÞ ¼ ~0 ¼ DðF Þx ðgð�x; xÞÞ. Sincegð�x; xÞ ¼ �gðx;�xÞ, we have DðF Þx ðgð�x; xÞÞ ¼ ~0 ¼ DðF Þx ðgðx;�xÞÞ. Thus, x 2 C*, and hence C � C*.

(ii) C� � C�1 is obvious.(iii) C�1 � C. Assume that x 2 C�1. Then x 2 K satisfies

DðF Þx ðgðx;�xÞÞ � DðF Þx ðgð�x; xÞÞ: ð1Þ

Suppose that x 2 C. Then F(x) F(y). By pseudo-invexity of �F, we have �DðF Þx ðgð�x; xÞÞ 2 @�. Since,

gðx;�xÞ ¼ �gð�x; xÞ, we have DðF Þx ðgðx;�xÞÞ 2 @�nf~0g. Using (1), we have, DðF Þx ðgðx;�xÞÞ 0 or DðF Þx ðgð�x; xÞÞ 0: In

970 S.K. Mishra et al. / European Journal of Operational Research 182 (2007) 965–970

view of Proposition 3.2, there exists a function p defined on K · K such that pðx;�xÞ 0, andF ðxÞ ¼ F ð�xÞ þ pðx;�xÞDðF Þx ðgðx;�xÞÞ � F ð�xÞ, a contradiction. Hence, x 2 C. h

Corollary 4.4. Let F be as in Corollary 4.2. If �x 2 C then then C ¼ C� ¼ C�1; where C� ¼ fx 2 K : DðF Þx ðx� �xÞ ¼DðF Þx ð�x� xÞg, and C�1 ¼ fx 2 K : DðF Þx ðx� �xÞ � DðF Þx ð�x� xÞg.

Proof. The proof of this corollary will follow from the proof of the Theorem 4.2, with gðx;�xÞ ¼ x� �x. h

5. Conclusion

Some results from Chew and Choo [4] and Ansari et al. [1] are extended to fuzzy mappings and using thesenew results an earlier work of Mangasarian [12] and Jeyakumar and Yang [7] concerning solution set of con-vex and pseudolinear programs are extended to pseudolinear and g-pseudolinear fuzzy programs.

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