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On fuzzy convexity of parametric programs
Mmroelectron Rehab, Vol 26, No 2, pp 235-239, 1986 Pnnted m Great Britain
0026-2714/8653 00+ 00 Pergamon Journals Ltd
ON FUZZY CONVEXITY OF PARAMETRIC PROGRAMS M. M. EL-KAFRAWY Inmtute of Natmnal Planning, Cmro, Egypt and N. A. EL-RAMLYand R. A. MAHMOUD Faculty of Smence, Menofia Umverslty, Egypt
(Recewedfor pubhcauon 1 August 1985) Abstract--The fuzzy sets is a notmn whmh generahzes all types of the existing sets Many authors were interested m this new type, so they defined many related notmns and mvestzgatedmore of thmr properties In this paper we study a new concept of fuzzinesscalled fuzzy convexity by introducing the notion of fuzzy convex sets for mathematical programs and also for parametric programs The connections between these notions and the ordinary ones are gwen and some of their characterizations are investigated INTRODUCTION
Throughout this work we shall use the notion of fuzzy sets to generalize some useful results which have been presented in the ordinary case by a great staff of researchers hke Rockafellar[18], No~l~ka and his followers [19,20] and others. Therefore this study presents a most qualitative study of mathematical programs which occurs In R" (the space of n-tuples of real numbers), even though many of the results can easily be formulated in a broader setting of functional analysis.
The concept of fuzzy sets was first defined in 1965 by Zadeh [1] as a generahzatlon of ordinary sets. Then many authors such as Goguen [2] m 1967, who introduced the notion of L-fuzzy and studied the algebraic properties. Also, C. L. Chang [3] in 1968 defined the fuzzy topological spaces as a generahzation of all previously known ones with some investigations of their characterizations Other apphcatlons of fuzzy sets had been introduced in 1970 when Foster[4] defined fuzzy topologmal group and studied some of FUZZY SETS its properties. But in 1971, Rosenfeld [5] studied the A subset F of R n called a fuzzy set if some sort of a properties of fuzzy groups notion The meaning of fuzzy uniform spaces was introduced by Lowen [6] in generalized "characteristic function" on R" whose 1974, also the concept of fuzzy vector spaces and fuzzy grades of membership may be more general than topological vector spaces were investigated and "yes" or "no" but It is characterized by a membership studied by Katsars and Llu [-7] in 1977 Recently in function. 1982, Mashour and Ghanlm [8] introduced the conPF'R" --~ M (1) cept of fuzzy closure spaces, then they studied many where M is the closed unit interval [0, 1]. properties of all the previous notions. The value I~F(x) at a point x e R " represents the Another dlrectmn of fuzzy apphcatlons started in 1965 when Zadeh introduced the notions of fuzzy grade of membership of x to F and so the best systems [9], fuzzy algorithms [-10, 11] and quantltaUve meaning of a fuzzy set is the ordered pair fuzzy semantics[12] and presented many of their {X,,UF(X)}VxsR". properties. While in 1971, S. K. Chang was interConsequently, if M = {0, 1}, then the fuzzy set ested in fuzzy programs [ 13], so he gave the theoretical coincides on the ordinary one. studies in this branch and some of its applications The fundamental mathematical operations of fuzzy Also fuzzy dynamic programming and the decision making processes[13] were presented by S. S. L. sets are defined as follows. Chang. In the same year K Asal and S Kitajlma Definiuon 1 [-14, 15] studied the important concepts which called For a collection {F,,zeI} of fuzzy subsets of R", the optimization control using fuzzy automata and a method for optimizing control of multimodel systems their union and intersection given by using the same previous notions. Many other studies kJ F , = x, U F,(x) , g x e R " (2) of fuzzy optimization had been done by some authors (1) tel IEI like Kandel and others [16, 17]. They presented the concept of minimization of fuzzy functions and its where comments, hence they investigated many of their t..JF,(x)=max{#e,(X),leI};VxsR n (3) properties IE] 235
{ #
}
M M EL-KAFRAWYet al
236
(2)
OF~=
x,~F,(x),VxeR"
IEI
Id
t
CONVEX
SETS
Defimtton 7
where F,(x)=mln
FUZZY
(4)
~el
t
OF,(x),teI
I. lel
,Vx6R"
(5)
)
Definmon 2 The distance between any two fuzzy subsets FI and F 2 of a finite set X (having n elements) will be called the Hamming distance and defined by
d(Fl, F2) = ~ [ # F I ( X t ) - - # F 2 ( X t ) [
(6)
z=l
A fuzzy subset A of R" is called fuzzy convex ff for each two points a 1 and a z of A having /~ 4(al) and # 4(a2), respectively and for each 7~ [0, 1] the fuzzy hne segment (Vii ÷ ( 1 - 7 )a2) ~ A having a membership function #* defined as kt* = 7FtI,,,&(I -7)PI,,:,
(14)
Remark 2 Since #* e [0, 1] and {0, 11 c [0, 1], this leads to the ordinary convex set being a special case of fuzzy convex set
Definmon 8
Definmon 3
A vector sum
The relanve Hamming distance 1s defined as ~(F1, F2) = 1-d(Fl,F2).
(7)
tl
t=l
is called a fuzzy convex combination of a,, ~ = 1,2, If
,m
Remark 1 a , ~ A V t = 1.2,
It is obvious from definition 3 that
0 <~ 6(F1.F z) <~ 1
,m, ~ 7, = 1 I--I
(8)
and all 7~ >~ 0 and having a membership function fi is given by
Definmon 4 For any two ordered n-tuples v = (kl,k 2. . . . k.) and v ' = (k'1,k~, .,k'.) such that each of k, and k',, t = 1, 2. , n belong to the same totally ordered set K, the dominates, strictly dominates relation between v and v' and will be denoted by v'~> v and v ' > v, respectively, and defined by
fi = Z 7,1J~,,,r
(151
i-1
The following theorem gives an equivalent definltion of fuzzy convex set.
Theorem 1
(9)
Let A be a convex set, then the following statements are equivalent
(10)
(1) A is a fuzzy convex set. (11) A contains all fuzzy convex combInanons of Its elements
The fuzzy graph of two sets A a and A 2 denoted by G 4, x A2(at, a2) for each ordered pair (a 1, az) s A t x A 2 and defined by
Proof (1) Let A be a fuzzy convex set, then the second statement is satisfied at m = 2 and for cases m > 2 can be easily proved by lnducnon Then (11) holds. (2) Let
v' >~ Vlffk',/> k,¥1 = 1,2 . . . . . n and
v' > Vlffk', > k, Vt = 1,2 . . . . n Definmon 5
G A,×4~(al, a2) = {(al,a2)l#6(al, a2)}
(11)
where #da~,a2) is the membership funcnon of each (al,a2 )~ A1 × A z and #G(al,a2)~ M.
a*= whenevera, e A V l =
Definition 6
A 2 =
1, 2.
,mind
~ ?,,= 1,
For any two fuzzy convex sets A 1 and A2, the direct sum of them denoted as A I [] A 2 and defined by
A1 []
~ 7,a,~A
l((al, a2),fiAa [] Az(ax, a2))] aa ~A1 andazeA2]
(12)
where
l--1
assume
7, 4: 1.re [1.2, then 1 - 2. # 0 and If we consider a'=
~ y',a, where l=l
fi 4,[]A2(al,a2) = l~6(al,a2)
(13)
,m~.
71-
1 -Vr'
Fuzzy sets
237
Defimtion 11
since m
The fuzzy convex hull of any fuzzy convex set A is the intersection of all fuzzy convex sets which contain A and denoted by f.c.H(A) i.e.
Z~:=I 1=1 t =/=r
then a' ~ A which leads to a* = 7,a, + ( 1 - 7 , ) a' belongs to A for each a , a' belongs to it and 7,/> 0. Then A IS a fuzzy convex set and this completes the proof.
Theorem 2 The Intersection of an arbitrary collection of fuzzy convex sets is a fuzzy convex set
Proof Let {A,, ~~ I~ be a collection of fuzzy convex sets, it is clear that O A , is also a fuzzy set (by defin7tion 1 and so ff we assume the two points a I and a2 m ~ A, this leads to that a x and a 2 belong to each A,, ze I and since each one of A,, t ~ 1 is a fuzzy convex set, then for any non-negative value ?, ~ [0, 1] the fuzzy lane segment )'al + ( 1 - ) , ) a 2 must belong to all A,, t e l and hence it belongs to O A, which gives tEI
the result.
Defimtton 9 The complement of any fuzzy subset A of R" denoted by A c and is defined as
A ~ = (x, 1 - p 4 ( x ) ) , V x ~ R "
(16)
fc.H(A) = O {A, [A = A, and A, IS fuzzy convex} (20) The following theorems present some characterizations of f.c.H(A).
Theorem 4 If A is a fuzzy convex set, then the following statements hold" (1) f.c.H(A) as a fuzzy convex set. 111) f.c.H(A) as the smallest fuzzy convex set contaming A. (ni) f.c.H(f.c.H(A)) = f.c.H(A). P r o o f (1) Follows directly by definition 11 and theorem 2. (7i) Since the intersection of some sets is smaller than each of them, then (n) is verified (liI) By using the previous two statements (7) and (il) we can immediately show the Third Statement.
Theorem 5 IfA 1 and A 2 are two fuzzy convex sets, the following hold (I) I f A 1 c A2, t h e n f c H(Aa) ~ f.c H(A2)
(21)
(17) fc.H(Aa)tJf.cH(A2) c f.c H(A a wA2).
(22)
(In) f . c . H ( A 1 ) ~ f . c . H ( A z ) ~ f c H ( A l r ~ A 2 )
(23)
where It A(X) ls the membership function of A
Remark 3 It is clear that A ~ is also a fuzzy set
Defimtton 10 I f A a and A2 are two fuzzy sets, so we can define the concepts.
Proof(l) By definition 1 1 we can easily write f.c h(Aa) = O
(1) The disjunctive sum A a 0) A2 defined by zl 1 ~ A 2 =
(A a ~ A~) u
( A 2 t~
A]).
× {A, I A, IS fuzzy convex 'q t E I and A a = A, I (17)
and
(77) The dTfference A x - A 2 = A1 n A~
(18)
f.c.H(A2) = O jEJ
x {Aj] Aj 7s fuzzy convex Vj ~ I and A 2 = Aj }
Remark 4 and since
F r o m (i) and (11) above, therefore
A 1@A 2 = (A1-A2)u(A2-AI).
(19)
Aa c A 2 ~ O {A, with hypothesis} IEl
Analogous to ordinary convexity and using definlt7ons 5, 6 and 10 we can investigate the following results without proof, since it is evident.
c O {Aj with ItS hypothesis} JEI
IfA a and A2 are any two fuzzy convex sets, then the following sets are fuzzy convex
which satisfies the first statement. (ii) Since A, c A 1 u A 2 , 7 = 1 , 2 by (i) we get directly the result. (ni) As the proof of the second one.
(I) The fuzzy graph G 4, × ~ (17) The &fference A , - A j , 1 = 1,2 a n d j = 2, 1 (in) The direct sum A, 03 Aj, t = 1,2 a n d j = 2, 1.
Corollary 1 f c H(A) consists of all fuzzy convex combinations of the elements of a fuzzy convex set A.
Theorem 3
26:2-C
M M EL-KAFRAWYet al
238
ProoJ. We showed i n (1) of theorem 4 that f.c.H(A) is a fuzzy convex and by the eqmvalent definition in theorem 1 complete directly the proof
and hence
Theorem 6
where
The image and inverse image of any fuzzy convex set under a blcontlnuous injection mapping are also fuzzy convex sets
Q(2,I) = {2 e A Ip(2,1lC~mov,(2) 4: O} (33)
mop,(2)={,YlF(~,2)= mIn F(x,)O 1 ~eMO)
(34)
)
and
Proof Follows directly by using the assumptions given in the theorem
p(ZI) =
{ x ~=G,(x,.~)=O,tEI.
FUZZY CHARACTERIZATIONS OF P A R A M E T R I C SETS
G,(x, 2)
(35)
l=l
Consider the parametric problem ~mln [F(x,2)Ix~M(2)),~A
(P;) [M().)= [xeXIG,(x,2) <~O,l~I}
(24) (25)
where the functions F(x,2) and G,(x,2), zel are non-linear convex functions (llnearaty is a special case) and A is the set of arbitrary real parameters Many authors lake No~l~ka et al in [19,20], Osman [21, 22], Berge [23] and others studied some special cases of problem (Pa) Also, many results of quahtatlve and quantitative studies are presented when parameters appear in the constraints, in the objective function and in both like our problem (Pa) The fundamental concepts of parameters are:
Definmon 12 (I) The constraint set mapping M as M.~ --* M(2).
126)
(n) The extreme value function 05 as 05:2-*0512)= mln F(x,;t)
(27)
",cM(2)
(llI) The optimal set mapping ~b as 2--.~,(2) = {xeM(R)lFlx,2)= 05(2)} (28) (iv) The set of feasible parameters ,~ = {2~AIM(2) 4:q51 (v) The solvability set ~ = {,~e A I q,(,~) ~ 051
(30)
x
G,(x,X)=O,t~Ic{1,2, .,k,], l
i=1
(1) ~ is non-empty, unbounded and convex If there is one parameter such that the set of constraints is bounded, then ~ is closed It is either one point or a cone with vertex at the origin d is a star shaped set with the zero element as its common point of visibility (2) o?/c ~. They are equal if the set of optimal solutions is bounded for at least one parameter with which the problem is solvable. If F(x,2) is strictly convex, then 4l is unbounded, convex and ~h' = (3) G(2) is closed and star shaped set with ~ as its common point of visibility (4) Q(2,I) have the property, if the objective functions are strictly convex and 2, 2* are two distinct values of parameters in A and corresponding to 7 and I*, respectively, with Q(f.,/-) 5a Q(2*, I*), then they are disjoint sets. By using fuzzy concepts we can generalize the previous results which stated without proof for it is obvious as
The set of feasible parameters is fuzzy convex if one of the following holds (1) If at least one parameter of A satisfies the boundness of the constraint set. (i1) If for all L d and non-negative parameters the sets M(21 ) and M(2 J ) are non-adjolnt.
(31)
(vii) The stability set of the second kind Q(2,I) if there exists ). with corresponding optimal point .~ such that Y • p(2-, I) where p(,~,I)=
Theorem 7
Theorem 8 (29)
(Vl) The stabihty set of the first kind with a corresponding optimal point 2 denoted by G(Y) as the following G(~) = [ 2 e A [F(2,)o) = minf(x,2)]
F r o m the these notions we conclude the following results.
Theorem 9 The solvability parametric set 4l is a fuzzy convex if any of the following conditions are satisfied (1) The objective function F(x, 2) contain a positive definite quadratic term. (li) The constraints set of the problem is bounded for at least one parameter where the problem IS solvable
Fuzzy sets REFERENCES
1 L A Zadeh, Fuzzy sets, l n f Control 8, 338-353 (1965) 2 J A Goguen, L-fuzzy sets, J math Analysts Apphc 18, 145 174 (1967) 3 C L Chang, Fuzzy topological spaces, J math Analysts Apphc 24, 182-190 (1968) 4 D H Foster, Fuzzy topologicalgroups, J math Analysts Apphc 67, 549-564 (1970). 5 A Rosenfeld, Fuzzy groups, J math Analysts Apphc 35, 512-517 (1971) 6 R Lowen, Topologies flous, C R Acad Scl Parts 278, 925 928 (1974) 7 A K Katsaras and D B. Llu, Fuzzy vectors spaces and fuzzy topological vectors spaces. J math Anolysts Apphc 58, 135-146 (1977) 8 A S Mashhour and M H. Ghanim, Fuzzy closure spaces, J math Analysts Appllc (1982) 9 L A Zadeh, Fuzzy sets and systems, Proceedmys of the Sympostum on Systems Theory. pp 29-39, Brooklyn Polytechnic Institute, Brooklyn, NY, (1966) 10 L A Zadeh, Fuzzy algorithms, Inf Control 12, 99-102 (19681 11 L A Zadeh, On fuzzy algorithms, E R L memo M 325, Umverslty of California, Berkeley, February 1972 12 L A Zadeh, Quantitative fuzzy semantics, Inf Sct 3 1 17 (1968) 13 S K Chang, Fuzzy programs, theory and apphcations, P I B Proc Comput Automata p. 21 (1971)
239
14 K Asai and S Kltajlma, Learning control ofmultlmodal systems of fuzzy automata, Pattern Recognmon and Machme Learnmg (Edited by K. S Fu) Plenum Press, New York (1971) 15 K Asai and S Kltajlma, A method for optimizing control of multlmodal systems using fuzzy automata, l n f Sct 3, 343-353 (1971) 16 A Kandel, On minimization of fuzzy functions. IEEE Trans Comput. C22 (1973) 17 A Kandel, Comments on the minimization of fuzzy functions, IEEE Trans Comput C22, 217 (1973) 18 R T. Rockafellar, Convex Analysts Princeton Umverslty Press (1969) 19 F No~l~Zka,J Guddat, H Hollatz and B Bank, Theorte der Lmearen Parametemschen OptimIerung Akademle, Berlin (1974) 20 B Bank, J Guddat, D Klatte, B Kummer and K Tammer, Non-Linear Parametr,c Opttmlzatton, Blrkhhuser, Basel-Boston-Stuttgart (1983) 21 M S A Osman, Qualitative analysis of basic notions m paramemc convex programming I (parameters in the constraints), Appl Math CSSR Akademtc red Praha 22, 318-332 (1977) 22 M S A Osman, A M Sarhan and A H. El-Banna, Stability of nonlinear programming problems with Multiparameters in the objective function Proc 1st Int. Conf France I, 175-179 (1981) 23 C Berge, Topologlcal Spaces Oliver and Boyd, London (1963)
0026-2714/8653 00+ 00 Pergamon Journals Ltd
ON FUZZY CONVEXITY OF PARAMETRIC PROGRAMS M. M. EL-KAFRAWY Inmtute of Natmnal Planning, Cmro, Egypt and N. A. EL-RAMLYand R. A. MAHMOUD Faculty of Smence, Menofia Umverslty, Egypt
(Recewedfor pubhcauon 1 August 1985) Abstract--The fuzzy sets is a notmn whmh generahzes all types of the existing sets Many authors were interested m this new type, so they defined many related notmns and mvestzgatedmore of thmr properties In this paper we study a new concept of fuzzinesscalled fuzzy convexity by introducing the notion of fuzzy convex sets for mathematical programs and also for parametric programs The connections between these notions and the ordinary ones are gwen and some of their characterizations are investigated INTRODUCTION
Throughout this work we shall use the notion of fuzzy sets to generalize some useful results which have been presented in the ordinary case by a great staff of researchers hke Rockafellar[18], No~l~ka and his followers [19,20] and others. Therefore this study presents a most qualitative study of mathematical programs which occurs In R" (the space of n-tuples of real numbers), even though many of the results can easily be formulated in a broader setting of functional analysis.
The concept of fuzzy sets was first defined in 1965 by Zadeh [1] as a generahzatlon of ordinary sets. Then many authors such as Goguen [2] m 1967, who introduced the notion of L-fuzzy and studied the algebraic properties. Also, C. L. Chang [3] in 1968 defined the fuzzy topological spaces as a generahzation of all previously known ones with some investigations of their characterizations Other apphcatlons of fuzzy sets had been introduced in 1970 when Foster[4] defined fuzzy topologmal group and studied some of FUZZY SETS its properties. But in 1971, Rosenfeld [5] studied the A subset F of R n called a fuzzy set if some sort of a properties of fuzzy groups notion The meaning of fuzzy uniform spaces was introduced by Lowen [6] in generalized "characteristic function" on R" whose 1974, also the concept of fuzzy vector spaces and fuzzy grades of membership may be more general than topological vector spaces were investigated and "yes" or "no" but It is characterized by a membership studied by Katsars and Llu [-7] in 1977 Recently in function. 1982, Mashour and Ghanlm [8] introduced the conPF'R" --~ M (1) cept of fuzzy closure spaces, then they studied many where M is the closed unit interval [0, 1]. properties of all the previous notions. The value I~F(x) at a point x e R " represents the Another dlrectmn of fuzzy apphcatlons started in 1965 when Zadeh introduced the notions of fuzzy grade of membership of x to F and so the best systems [9], fuzzy algorithms [-10, 11] and quantltaUve meaning of a fuzzy set is the ordered pair fuzzy semantics[12] and presented many of their {X,,UF(X)}VxsR". properties. While in 1971, S. K. Chang was interConsequently, if M = {0, 1}, then the fuzzy set ested in fuzzy programs [ 13], so he gave the theoretical coincides on the ordinary one. studies in this branch and some of its applications The fundamental mathematical operations of fuzzy Also fuzzy dynamic programming and the decision making processes[13] were presented by S. S. L. sets are defined as follows. Chang. In the same year K Asal and S Kitajlma Definiuon 1 [-14, 15] studied the important concepts which called For a collection {F,,zeI} of fuzzy subsets of R", the optimization control using fuzzy automata and a method for optimizing control of multimodel systems their union and intersection given by using the same previous notions. Many other studies kJ F , = x, U F,(x) , g x e R " (2) of fuzzy optimization had been done by some authors (1) tel IEI like Kandel and others [16, 17]. They presented the concept of minimization of fuzzy functions and its where comments, hence they investigated many of their t..JF,(x)=max{#e,(X),leI};VxsR n (3) properties IE] 235
{ #
}
M M EL-KAFRAWYet al
236
(2)
OF~=
x,~F,(x),VxeR"
IEI
Id
t
CONVEX
SETS
Defimtton 7
where F,(x)=mln
FUZZY
(4)
~el
t
OF,(x),teI
I. lel
,Vx6R"
(5)
)
Definmon 2 The distance between any two fuzzy subsets FI and F 2 of a finite set X (having n elements) will be called the Hamming distance and defined by
d(Fl, F2) = ~ [ # F I ( X t ) - - # F 2 ( X t ) [
(6)
z=l
A fuzzy subset A of R" is called fuzzy convex ff for each two points a 1 and a z of A having /~ 4(al) and # 4(a2), respectively and for each 7~ [0, 1] the fuzzy hne segment (Vii ÷ ( 1 - 7 )a2) ~ A having a membership function #* defined as kt* = 7FtI,,,&(I -7)PI,,:,
(14)
Remark 2 Since #* e [0, 1] and {0, 11 c [0, 1], this leads to the ordinary convex set being a special case of fuzzy convex set
Definmon 8
Definmon 3
A vector sum
The relanve Hamming distance 1s defined as ~(F1, F2) = 1-d(Fl,F2).
(7)
tl
t=l
is called a fuzzy convex combination of a,, ~ = 1,2, If
,m
Remark 1 a , ~ A V t = 1.2,
It is obvious from definition 3 that
0 <~ 6(F1.F z) <~ 1
,m, ~ 7, = 1 I--I
(8)
and all 7~ >~ 0 and having a membership function fi is given by
Definmon 4 For any two ordered n-tuples v = (kl,k 2. . . . k.) and v ' = (k'1,k~, .,k'.) such that each of k, and k',, t = 1, 2. , n belong to the same totally ordered set K, the dominates, strictly dominates relation between v and v' and will be denoted by v'~> v and v ' > v, respectively, and defined by
fi = Z 7,1J~,,,r
(151
i-1
The following theorem gives an equivalent definltion of fuzzy convex set.
Theorem 1
(9)
Let A be a convex set, then the following statements are equivalent
(10)
(1) A is a fuzzy convex set. (11) A contains all fuzzy convex combInanons of Its elements
The fuzzy graph of two sets A a and A 2 denoted by G 4, x A2(at, a2) for each ordered pair (a 1, az) s A t x A 2 and defined by
Proof (1) Let A be a fuzzy convex set, then the second statement is satisfied at m = 2 and for cases m > 2 can be easily proved by lnducnon Then (11) holds. (2) Let
v' >~ Vlffk',/> k,¥1 = 1,2 . . . . . n and
v' > Vlffk', > k, Vt = 1,2 . . . . n Definmon 5
G A,×4~(al, a2) = {(al,a2)l#6(al, a2)}
(11)
where #da~,a2) is the membership funcnon of each (al,a2 )~ A1 × A z and #G(al,a2)~ M.
a*= whenevera, e A V l =
Definition 6
A 2 =
1, 2.
,mind
~ ?,,= 1,
For any two fuzzy convex sets A 1 and A2, the direct sum of them denoted as A I [] A 2 and defined by
A1 []
~ 7,a,~A
l((al, a2),fiAa [] Az(ax, a2))] aa ~A1 andazeA2]
(12)
where
l--1
assume
7, 4: 1.re [1.2, then 1 - 2. # 0 and If we consider a'=
~ y',a, where l=l
fi 4,[]A2(al,a2) = l~6(al,a2)
(13)
,m~.
71-
1 -Vr'
Fuzzy sets
237
Defimtion 11
since m
The fuzzy convex hull of any fuzzy convex set A is the intersection of all fuzzy convex sets which contain A and denoted by f.c.H(A) i.e.
Z~:=I 1=1 t =/=r
then a' ~ A which leads to a* = 7,a, + ( 1 - 7 , ) a' belongs to A for each a , a' belongs to it and 7,/> 0. Then A IS a fuzzy convex set and this completes the proof.
Theorem 2 The Intersection of an arbitrary collection of fuzzy convex sets is a fuzzy convex set
Proof Let {A,, ~~ I~ be a collection of fuzzy convex sets, it is clear that O A , is also a fuzzy set (by defin7tion 1 and so ff we assume the two points a I and a2 m ~ A, this leads to that a x and a 2 belong to each A,, ze I and since each one of A,, t ~ 1 is a fuzzy convex set, then for any non-negative value ?, ~ [0, 1] the fuzzy lane segment )'al + ( 1 - ) , ) a 2 must belong to all A,, t e l and hence it belongs to O A, which gives tEI
the result.
Defimtton 9 The complement of any fuzzy subset A of R" denoted by A c and is defined as
A ~ = (x, 1 - p 4 ( x ) ) , V x ~ R "
(16)
fc.H(A) = O {A, [A = A, and A, IS fuzzy convex} (20) The following theorems present some characterizations of f.c.H(A).
Theorem 4 If A is a fuzzy convex set, then the following statements hold" (1) f.c.H(A) as a fuzzy convex set. 111) f.c.H(A) as the smallest fuzzy convex set contaming A. (ni) f.c.H(f.c.H(A)) = f.c.H(A). P r o o f (1) Follows directly by definition 11 and theorem 2. (7i) Since the intersection of some sets is smaller than each of them, then (n) is verified (liI) By using the previous two statements (7) and (il) we can immediately show the Third Statement.
Theorem 5 IfA 1 and A 2 are two fuzzy convex sets, the following hold (I) I f A 1 c A2, t h e n f c H(Aa) ~ f.c H(A2)
(21)
(17) fc.H(Aa)tJf.cH(A2) c f.c H(A a wA2).
(22)
(In) f . c . H ( A 1 ) ~ f . c . H ( A z ) ~ f c H ( A l r ~ A 2 )
(23)
where It A(X) ls the membership function of A
Remark 3 It is clear that A ~ is also a fuzzy set
Defimtton 10 I f A a and A2 are two fuzzy sets, so we can define the concepts.
Proof(l) By definition 1 1 we can easily write f.c h(Aa) = O
(1) The disjunctive sum A a 0) A2 defined by zl 1 ~ A 2 =
(A a ~ A~) u
( A 2 t~
A]).
× {A, I A, IS fuzzy convex 'q t E I and A a = A, I (17)
and
(77) The dTfference A x - A 2 = A1 n A~
(18)
f.c.H(A2) = O jEJ
x {Aj] Aj 7s fuzzy convex Vj ~ I and A 2 = Aj }
Remark 4 and since
F r o m (i) and (11) above, therefore
A 1@A 2 = (A1-A2)u(A2-AI).
(19)
Aa c A 2 ~ O {A, with hypothesis} IEl
Analogous to ordinary convexity and using definlt7ons 5, 6 and 10 we can investigate the following results without proof, since it is evident.
c O {Aj with ItS hypothesis} JEI
IfA a and A2 are any two fuzzy convex sets, then the following sets are fuzzy convex
which satisfies the first statement. (ii) Since A, c A 1 u A 2 , 7 = 1 , 2 by (i) we get directly the result. (ni) As the proof of the second one.
(I) The fuzzy graph G 4, × ~ (17) The &fference A , - A j , 1 = 1,2 a n d j = 2, 1 (in) The direct sum A, 03 Aj, t = 1,2 a n d j = 2, 1.
Corollary 1 f c H(A) consists of all fuzzy convex combinations of the elements of a fuzzy convex set A.
Theorem 3
26:2-C
M M EL-KAFRAWYet al
238
ProoJ. We showed i n (1) of theorem 4 that f.c.H(A) is a fuzzy convex and by the eqmvalent definition in theorem 1 complete directly the proof
and hence
Theorem 6
where
The image and inverse image of any fuzzy convex set under a blcontlnuous injection mapping are also fuzzy convex sets
Q(2,I) = {2 e A Ip(2,1lC~mov,(2) 4: O} (33)
mop,(2)={,YlF(~,2)= mIn F(x,)O 1 ~eMO)
(34)
)
and
Proof Follows directly by using the assumptions given in the theorem
p(ZI) =
{ x ~=G,(x,.~)=O,tEI.
FUZZY CHARACTERIZATIONS OF P A R A M E T R I C SETS
G,(x, 2)
(35)
l=l
Consider the parametric problem ~mln [F(x,2)Ix~M(2)),~A
(P;) [M().)= [xeXIG,(x,2) <~O,l~I}
(24) (25)
where the functions F(x,2) and G,(x,2), zel are non-linear convex functions (llnearaty is a special case) and A is the set of arbitrary real parameters Many authors lake No~l~ka et al in [19,20], Osman [21, 22], Berge [23] and others studied some special cases of problem (Pa) Also, many results of quahtatlve and quantitative studies are presented when parameters appear in the constraints, in the objective function and in both like our problem (Pa) The fundamental concepts of parameters are:
Definmon 12 (I) The constraint set mapping M as M.~ --* M(2).
126)
(n) The extreme value function 05 as 05:2-*0512)= mln F(x,;t)
(27)
",cM(2)
(llI) The optimal set mapping ~b as 2--.~,(2) = {xeM(R)lFlx,2)= 05(2)} (28) (iv) The set of feasible parameters ,~ = {2~AIM(2) 4:q51 (v) The solvability set ~ = {,~e A I q,(,~) ~ 051
(30)
x
G,(x,X)=O,t~Ic{1,2, .,k,], l
i=1
(1) ~ is non-empty, unbounded and convex If there is one parameter such that the set of constraints is bounded, then ~ is closed It is either one point or a cone with vertex at the origin d is a star shaped set with the zero element as its common point of visibility (2) o?/c ~. They are equal if the set of optimal solutions is bounded for at least one parameter with which the problem is solvable. If F(x,2) is strictly convex, then 4l is unbounded, convex and ~h' = (3) G(2) is closed and star shaped set with ~ as its common point of visibility (4) Q(2,I) have the property, if the objective functions are strictly convex and 2, 2* are two distinct values of parameters in A and corresponding to 7 and I*, respectively, with Q(f.,/-) 5a Q(2*, I*), then they are disjoint sets. By using fuzzy concepts we can generalize the previous results which stated without proof for it is obvious as
The set of feasible parameters is fuzzy convex if one of the following holds (1) If at least one parameter of A satisfies the boundness of the constraint set. (i1) If for all L d and non-negative parameters the sets M(21 ) and M(2 J ) are non-adjolnt.
(31)
(vii) The stability set of the second kind Q(2,I) if there exists ). with corresponding optimal point .~ such that Y • p(2-, I) where p(,~,I)=
Theorem 7
Theorem 8 (29)
(Vl) The stabihty set of the first kind with a corresponding optimal point 2 denoted by G(Y) as the following G(~) = [ 2 e A [F(2,)o) = minf(x,2)]
F r o m the these notions we conclude the following results.
Theorem 9 The solvability parametric set 4l is a fuzzy convex if any of the following conditions are satisfied (1) The objective function F(x, 2) contain a positive definite quadratic term. (li) The constraints set of the problem is bounded for at least one parameter where the problem IS solvable
Fuzzy sets REFERENCES
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