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Stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints
FUZZY
sets and systems Fuzzy Sets and Systems74 (1995) 343-351
ELSEVIER
Stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints M o h a m e d Abd E1-Hady Kassem*, Elsaid Ibraim A m m a r Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
ReceivedJanuary 1994;revisedJuly 1994
Abstract
This paper deals with the stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraint functions.These fuzzyparameters are characterized by fuzzynumbers. Qualitative and quantitative analysis of the basic notions like the set of feasible parameters, the solvability set, the stability sets of the first kind and of the second kind, will be reformulated under the concept of ct-pareto optimality. An illustrative example is given to clarify the obtained results. Keywords: Multiobjectivenonlinear programming; Stability;Parametric analysis;Fuzzy programming; Fuzzy numbers;
7-Pareto optimality
1. Introduction
Some basic notions like the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind for parametric convex nonlinear programming problems were introduced by Osman [6]. Tanaka and Asai [11] formulated multiobjective linear programming problems with fuzzy parameter. Orlovski [5] formulated a general multiobjective nonlinear programming problem with fuzzy parameters. Sakawa and Yano [10] introduced the concept of ~-pareto optimality of multiobjective nonlinear programming problem with fuzzy parameters. In this paper, qualitative and quantitative analysis of the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind for multiobjective nonlinear programming problem with fuzzy parameters in the constraints are studied by using the concept of at-pareto optimality. Also an algorithm for obtaining the stability set of the first kind which corresponds to ~-pareto optimal solution is introduced.
*Corresponding author. 0165-0114/95/$09.50 © 1995- ElsevierScienceB.V.All rights reserved SSDI 0165-01 14(94)00344-0
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M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
2. Problem formulation Let us consider the following multiobjective nonlinear programming problem with fuzzy parameters in the constraints
P~,b:
minimize F(x) subject to x ~ X(b, ~) = {x ~ ~" I G(x, ~) <<.b},
where F : ~ " ~ R m, G : ~ " ~ R " are convex functions of class C (1) on ~", F = ( f l , f 2 . . . . . f~)T, G = (gx, g2, ".', g,)T, (bl, b2 .... , b,) T represent the vector of real numbers and ~ = (vl, v2,--., ~7~)represent a vector of fuzzy parameter in the constraints gj(x,~j), j = 1,2,...,r. These fuzzy parameters are assumed to be characterized by fuzzy numbers. The fuzzy numbers ~ form a convex continuous fuzzy subset of the real line whose membership function i~(v t) (t = 1,2, 3,4) is defined by [101: 1. a continuous mapping from ~ to the closed interval [0, 1]; 2. /~(v) = 0 for all v e ( - 0% vl]; 3. strict increase on Iv ~, v21; 4. /t~(v) = 1 for all v ~ Iv 2, v3]; 5. strict decrease on Iv 3, v*]; 6. /~(v) = 0 for all v ~ [v 4, ~). Here, we assume that the membership function/~(v) is differentiable on Iv x, v*] . Remark 1. It must be noted that if/~(v) is a concave function on [v 1, v*], the conditions (1)-(6) which are imposed on /~(v) are still satisfied and the membership function/J~(v) is a function of vt, t = 1, 2, 3, 4 at a certain level a. Definition 1. The a-level set of the fuzzy numbers ~j is defined as the ordinary set L~(q) for which the degree of their membership functions exceeds the level a: L~(~) = {vlp~j(vj) >~ a, j = 1, 2 . . . . . r}. For a certain degree a, the problem P~.b can be written in the following nonfuzzy form 1-91: aM:
minimize
F(x)
subject to
N(b, v') = {(x, v) E R"+rlgj(x, vj) ~ bj, j = 1, 2 , . . . , r; vj~ L~(~j), j = 1, 2 .... , r},
In the whole paper, we assume that the problem aM is stable [9]. Definition 2. x* ~ X(b, ~) is said to be an a-pareto optimal solution to the problem aM, if and only if there does not exist another x ~ X (b, ~), v ~ L~(q) such that F (x) ~ F(x* ), and F(x) ~ F(x*), where the corresponding values of parameters v* are called a-level optimal parameters. We see that the a-pareto optimal solutions of the problem aM can be characterized in terms of the a-optimal solution of the kth-objective e-constraint problem aM, [21, which takes the following form: aM~:
minimize
fk(X)
subject to
S,(b, ¢) = {(x, v) e R"+r IOj(X,vi) <<.b~, j = 1, 2 , . . . , r, vj ~ L~(~j), j = 1, 2,..., r, f~(x)~
where e = (el,e2, ... , e k - l , e k + l , ... ,era) T.
M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
345
Theorem 1. x* is an a-pareto optimal solution of problem aM if and only if x* solves ~M~ for every k =
1,...,m. Proof. Necessity. Assume that (x*, v*) does not solve aM,. for some k, then there exists (x, v) ~ NAb, v t) such thatfk(x) < fk(X*) implies that (x*, v*)~ N* (N* is the set of all a-pareto optimal solutions of the problem aM). Thus the conclusion follows. Sufficiency. Since (x*, v*) solves the problem aM,. for e = e* and every k = 1, 2 .... , m, then there is no other (x, v) ~ NMb, v t) such that f a x ) <~f~(x*), s = 1, 2 , . . . , m with strict inequality holding for at least one s. This implies by definition that (x*, v*) e N*. []
3. The set of feasible parameters Definition 3. (I) The set of feasible parameters for problem aM is denoted by U and defined by
U = {(b,v t) ~ ~5"[N(b,v') ~ 0}, (II) The set of feasible parameters for the problem aM~ is denoted by U~ and defined by U,: = {(e,b,v') e ~m+5r-'lN~(b,v') ~ 0}. Theorem 2. The sets U and U~ are convex.
Proof. T o prove the set U is convex, assume that (b 1, vtl), (b 2, V/2) are two points in U (v a = (v ~, ~)2i,v3i V4i), • llltl i = 1,2), then there exist the a-pareto optimal solutions x 1 and x 2 such that gj(x~,v~ 1) <<.b~, I~j~ j ) >>,a and ~j(X 2 ' yr.2 2 #~(v~2)/> a, j -- 1, 2 . . . . . k. Then for all 0 ~ 2 ~< 1 we have (1 - 2) Oj(x ~, v~~) + 2Oj(x 2, v~2) ~< j ) ~< b~, (1 - 2)b) + Abe, and (1 - 2")p~j~"vjt~") + 2 /%~,~.t2, ~ ~ >/(1 - 2)a + 2a = a. Since gj are convex functions for each j in x and/~:(v:) are concave membership functions in v, then for all 0 ~< 2 ~< 1 we have gj[(1 - 2)(x ~, v~x) + 2(x2,v~2)] ~< (1 - 2)b~ + 2b], j = 1,2 . . . . . r; and p~((1 - 2)v~ ~ + 2v~2)/> a, j = 1,2 ..... r. Then N[(1 - 2)b ~ +2b 2,(1-2)v '~+2v a]~0, i.e. [ ( 1 - 2 ) b x + 2 b z , ( 1 - 2 ) v " + 2 v ' 2 ] e U for all 0~<2~<1. Hence U is convex set. Similarly, we can prove that U~ is convex set. [] Theorem 3. I f gj(x, vj) and F(x) are lower semicontinuous for all (x, v), and t%(vj) are upper semicontinuous for
all v, then the sets U and U, are closed. Proof. Let us consider the point-to-set mapping F: U ~ R 5r which assigns a subset (b, v') ~ F(b, v ~) to each element (b, v') e U. Consider the sequences (b t"), v "")) e U, and (x t"), v "")) ~ R n+4" such that (x ~"),v "")) e F(b ~"~, v"")), gj(x ("), v~(")) <<.b~"), #~j(V~(n)) ~ ~ and b (") ~ b t°), (x ("), v "")) ~ (x (°), v"°)), v "") ~ v "°) as n -+ oo. F r o m the assumption, we have gj(x ~°), v~(°)) <~ limn-~ ® ~J, ,,.tx ~), v~t~)) ~< ~i, ho ~. j ~~.,(o)~ J J = lim,_,o~ p~j(v~t~)) >~ a. Then (x t°), v t°)) N(b ~°), v(°)), i.e., N(b (°), v ~°)) ~ O. Hence (b t°), v t°)) ~ U. Therefore U is closed. Similarly, we can prove that U, is closed. [] 4. The solvability set Definition 4. (I) The solvability set of problem a M is denoted by B and defined by
B = {(b,v') e Ulm,_opt(b,v')50}, where m~_op,(b, v') = {(if, 9)~ ~"+'13 a-optimal solution to the problem aM}
M.A.E. Kassem, E.I. Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
346
(II) The solvability set of problem ~M~ is denoted by B, and defined by B~ = { (e,b, v') • U~l m~.ovt(e,b, v') ~ 0 } ,
where m~.opt(e,b,v') = {(x,v)~ ~"+rl q ~-optimal solution to the problem ctM,}. Theorem 4. (I) I f for one point (b, v t) ~ B such that the set m~_opt(b,v t) is bounded, then B = U. (II) I f for one point (e, b, v ~) ~ B, such that the set m~_opt(e,b, v ~) is bounded, then B, = U~. Proof. (I) From the assumption and [6], we have the sets A1 = {(e, b, vt) 6 Rm+4r-11N(e, b, v t) ~ 0}, and
A2 = { (e, b, vt) • A11 problem ~M~ has ~-pareto optimal solution} are equal. Since the projection of A~ and A2 in 5r-space are equivalent, B = U. Similarly, we can prove (II). [] Lemma 1. Under the assumption of Theorem 4, the sets B and B~ are convex.
The proof follows from Theorems 4 and 2. Lemma 2. I f the set B v~ 0 (B~ vL O) and either F (x) ( fk(X) ) is strictly convex on ~n or the set N (b, v') (N~(b, v') ) is bounded for one (b,v t) • U ((e,b,v t) • U,), then B = U (B~ = U,).
The proof follows directly from Theorem 4. Theorem 5. (I) The problem ~M has an ot-pareto optimal solution Y~ if and only if F(x) has a subgradient (b,v t) at (~,~), (b, v t ) • B such that b(x-~)>>.O, b(v-f)>>.O, v t ( x - ~ ) > ~ O , and v t ( v - f ) > ~ O for all (x, v) • N(b, vt). (II) The problem ~M, has an ~-optimal solution (~, ~) if and only if fk(X) has a subgradient (e, b, vt) at (,2, ~), (e, b, v t) e B, such that e(x - ~) >~ O, e(v - f) >1 O, b(x - Y~) >>,O, b(v - ~) >>.O, vt(x - ~) >1 0 and v'(v - ~) >_,0 for all (x, v) • N~(b, v').
The proof is similar to the one in [6].
5. The stability set of the first kind Definition 5. (I) Suppose that (b, v t) • B with a corresponding ~t-pareto optimal solution ~ and the or-level
optimal parameter q of the problem ~M. Then the stability set of the first kind corresponding to (2, q) which is denoted by G(~, f), is defined by G(~, ~) = {(b, vt) • B I(~, ~) is an ~t-pareto optimal solution of the problem ctM}. (II) Suppose that (g, b, qt) • B, with a corresponding ~-optimal solution (~, q) of the problem ~M,, then the stability set of the first kind for the problem ctM, is Q(~, q) = {(e, b, vt) ~ B, I(x, q) is an 0t-optimal solution of the problem ~M,}. Lemma 3. I f the set M is defined as
M = {(b, v') • B[gj(~, ~ ) <%bj, I%(~)/> ~, J = 1, 2..... r; t = 1, 2, 3, 4}, then it is equal to the set ~1 where
A4 - {(b, v t) e B] 3 e such that minfk(x) ~
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M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
Proof. It follows from the fact that (Y, ~) e N(b, v'), where ~ is an ~-pareto optimal solution and ~ is the or-level optimal parameter of the problem ~M, for every (b, v') • M (and from [5]) that the set
= {(e,b,v') • B~[fs(Y) ~< es, s = 1,2 . . . . . m, s # k, Oj(ff, 9~) ~< bj,/~j(v~) >/~,j = 1,2, ... ,r;t = 1,2,3,4}, and the set
]fl = {(e,b,v')• B~,
min
(x, v)eN,~(b,vt)
fk(X' <~fk(X)}
are equal. Since the projections of the above sets A~t in 5r-space are equivalent, the result follows.
[]
R e m a r k 2. F r o m L e m m a 3, it follows that G(~, 9) _ M. T h e o r e m 6. The sets G(F,,9) and G~(F,,~) are star shaped with common point of visibility b i = 9j(~,gj),
= #~j(9~), j = 1, 2,..., r. Proof. Since (b, ~') is c o m m o n point of visibility of G(~, ~), i.e., Oj(~, fj) = bj, /z~j(9~) = ~, j = 1, 2 . . . . . r;
t = 1,2,3,4. Assume that (b°,v °') • G(~,9), then there exists an index set J _ {1,2,...,r} such that gj(~, 9~) = b °, #~j("vj°'') = a, j • Jo; gj(x, 9j) <~ b °, # ~jvj ( .o,,j > ~ & j • { 1 , 2 , . . . , r } - J o . F r o m the convexity of g)(x, vj) and concavity of/%(vj) we get for0~<2~
9j[(l-2)x
1+2x z,(1-2)v)
+ 2 v 2]
~< (1 - ~.)gi(xl,vl) + 2gj(x2, v]),
(1)
zvt2x #~j[(1 - 2)v~ ~ + 2v~2 ] / > (1 - 2)/~,(v~ ~) + /~/a~,~ j j.
(2)
F r o m the assumption, we have the following two equations for j • Jo; (1 -- 2)9j(~, ~ ) + ~gj(x,- ovy) -- (1 - 2)b~ + 2b ° = /gj, (1 - 2 ) # ~ ( ~ ) + 2#~(v °') = (1 -- 2)~ + 2~ = a,
for j • Jo,
j • So;
(3) (4)
and the following two equations for JCJo: (1 - 2)gj(ff, f~) + 29~(g,v °) <-%(1 - 2)bj + 2b ° = b~, for jCJo,
(5)
(1 - 2)/~(f~) + 2#~(v °') ~> (1 - 2)~ + ~.0~= ~,
(6)
for jCJo.
F r o m (1), (3), (5) we get
gj[2x' -t- (1 - 2)x z, 2v~ + (1 - 2)v~] = O)(~,vS) = b), j • Jo; gj[2x a + (1 - 2)x z, 2v) + (1 - 2)v~] = gj(~, f)) ~< b), j • Jo, where :~ = 2x ~ + (1 - 2 ) x z, f ) = 2 v ) ' + (1 - 2 ) v ~ ' . Also from the equations (2), (4), (6) we get / ~ ( ~ j ) = ~, for j • Jo, /~)(f~)/> ~, for JCJo, where ~' = 2f' + (1 - 2)v °'. Then the point (b, ¢') = [(1 - 2)b + 2b °, (1 - 2)9 t + 2v *t] • G(ff,,7). Hence the result. [] Similarly, we can prove that the set G,(~, ~) is star shaped.
M.A.E. Kassem, E.I. Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
348
6. Determination of the stability set of the first kind
Let (b, qt) • B with an a-pareto optimal solution £, the corresponding Karush-Kuhn-Tucker conditions will then have the following form:
~(~, ~j) ~fk(£) (~f~(2) Ox---~-+ ~ v , ~ - x p + L uj - - - 0 , i =s
j=1
(~x~
fl = 1, 2,...,n,
(7)
.....
(8)
k#s
~g~(~,~j)
3
~(~j)
~V0
j=1
~Y0
j= 1
f,(£)~
r,
s = l , 2 ..... m, s # k ,
(9)
gj(£, ~j) <~ bj, j = 1, 2 ..... r,
(10)
/%(~j) t> ct, j = 1,2,...,r,
(11)
vs[fs(£) - e~] = 0,
(12)
Uj[gj(X, f j ) --
bj]
s = 1,2 .... ,m; s # k,
=0,
j = 1,2 ..... r,
(13)
6j[#~,(~Tj) - ~] = 0, j = 1,2 ..... r,
(14)
v~>0,
(15)
s = l , 2 ..... m, s v~ k,
uj >~ O, 6j >~ O,
j = 1 , 2 ..... r,
(16)
From the Eqs. (7), (8), (15) and (16), we can determine the value of uj and 6j. According to these values, the stability set of the first kind G(£, q) will be determined as follows: (1) I f u j = O , J j = O , j = l , 2 ..... r, GI(£, f) = {(b,v')lgj(£, ~j) ~< bj, t%(~})/> a}(2) I f u j > O , 6 j > O , j =
1,2 ..... r,
Gz(£,q) = { ( b , v ' ) l g j ( £ , f j ) = bj, t%(f}) = ~}. (3) I f u j > O , 6 j = O , j = l , 2
bj>0, j=
(18)
..... r,
Ga(£, ~) = {(b, v')lgj(£, 5 ) = bj, #~(q))/> a}. (4) I f u j = 0 ,
(17)
(19)
1,2, .... r,
64(£, f) = {(b, vt)lg~(£, qj) <~ bj; #~(ft) = a}.
(20)
(5) If uj = O, t~j = O, j • J ~ {1,2 ..... r}, and uj > O, t~j > O, jq~J, G~(£, f) = {(b, v')I gj(£, fj) <~ b j , / % ( ~ ) >/a; and gj(£, ~j) = bj,/~(q~) = a}.
(21)
Gs(£,q).=
(22)
Thus U
a,(£,~).
possible J
The algorithm
(1) For a certain a • [0, 1], we select an arbitrary (b,~')• B and solve the problem aM to obtain an :t-pareto optimal solution ~. (2) Formulate the K a r u s h - K u h n - T u c k e r conditions (1)-(10).
M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343-351
349
(3) From the Eqs. (7), (8), (15) and (18), we determine the values of u~ and 6j by using a suitable algorithm. (4) According to the values of u~ and 6~, the stability set of the first kind G(~, f) can be determined as in the Eqs. (17)-(22).
7. The stability set of the second kind Definition 6. (I) We suppose that (b, ~t) ~ B with a corresponding a-pareto optimal solution ~ of the problem aM such that (£,~) ~ a(b, qt;J), where
a(b, q'; J) = {(x, v) ~ E"+r Lgj(x, vj) = [~j, I~j(q)) = a, j ~ J c {1, 2..... r} and
gj(x, vj) <~ bj, la~j(9)) >~ a, jCJ}. Then, the stability set of the second kind corresponding to a(b, 9t; j ) denoted by Q(a(b, ~t; j)) is defined by
Q(a(b,~';J)) = {(b, v') e BIE(b,v')c~a(b,9';J)
# 0},
where E(b, v') denotes the set of all a-pareto optimal solutions of problem aM. (II) We suppose that (g,b,9')~B~ with a corresponding a-optimal solution Y of ~M~ where (y, ~7)~ a~(b, 9'; J), where a~(b,9'; J ) = {(x,v) ~ R"+r I f~(x)= g~, S ~ I = {1,2, ... ,m}, s ¢ k,
gj(x, vj) = [gj, p~j(q)) = a, j 6 J = {1,2,...,r} and L(x) <~ g~, s¢l, s ~ k, gj(x, vj) <~ bj, #~j(9)) >1 a, jCJ}. Then, the stability set of the second kind corresponding to try(b, 9'; J) denoted by Q(ae(b, ~'; J)) is defined by
O~(ae(b, 9'; J)) = {(~, b, v') ~ B~ I E(~, b, v') c~ ae(b, 9'; J) 4: 0}, where E(e, b, v t) denotes the set of all 0t-optimal solutions of problem aM~. Remark 3. From the definitions of the stability sets of the first kind and of the second kind, we have (I) Q(a({9, 9'; J ) ) = U ~* G( xi, v~), where I = { i l (x i, v i) ~ a(b, ~'; J) and x i is an a-pareto optimal solution of the problem aM}. (II) Q~(a~([~,~';J))= Ui~1G(x~,vi), where I = {i[(xi, vi)~ae(b, gt;J) and x i is an a-optimal solution of problem the aM~}. Theorem 7. (I) I f the functions J~(x), i = 1,2, ...,m are strictly convex on ~" and (b 1, ¢1), (b2, v,2) are two
distinct points in B with Q(a(bl, vtl; J~)) 4: Q(a(b2,vt2; J2)). Then Q(a(b 1, vt~; J1 )) 0 Q(a(b 2, v '2, J2)) :/: 0. (II) If the function fk(x) is strictly convex on ~" and (e, b ~, v'~), (e, b 2, v t2) are two distinct points in B~ with Q,(a~(ba,vt~;J~)) ve Q~(a~(b2, v'2;J2)). Then
Q~(a~(b~, vtl; J~))~Q~(a,(b2, vt2; J2)) ¢ O. Proof. 0) By the assumption, it follows that J ~ J 2 . Suppose that Q(a(b 2, v'2; J2)) then there exists an a-pareto optimal solution if, such that (Y, 9) e tr(b, "7';J~ ) & a(b, ~'; Ja),
([~,f*)~Q(a(b~,vta;J~))~
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M.A.E. Kassem, E.I. A m m a r / Fuzzy Sets and Systems 74 (1995) 3 4 3 - 3 5 1
where
a(b,~';J1) = {(x,v) e R"+'lo~(x,v~) = b~, #~j(j) ~' = ~ , J ~ J 1 ; g j ( x , v j ) < D j , l ~ j ( ¢ J ) > ~ , J C J a } , tr([~, U; Jz) = {(x, v) ~ ~"+" [gj(x, vj) = b j, #~s(~) = ~, j 6 J2; gj(x, vj) < bj, p~j(~}) > 0~,JqsJ2 }. Therefore, 9s(x,v~) = b~, l~o,(g~) = ~; 9~(x, vs) < b~,/~s(g~) > ~, for at least one s • {1,2 . . . . . r} if J1 ¢ Jz, and then J1 = Jz which is contradiction. Hence the result follows. Similarly, we can prove (II). [] Example. Consider the following fuzzy problem: P{~.b):
minimize
(X 2 -- X2, X 2 At- X 2 )
subject to
~lX~ + ~2x2 ~< bl, 2~71x~ - 5~2x2 ~< b2.
The equivalent nonfuzzy problem takes the form: eM:
minimize
(x 2 - x 2 , x 2 + x 2)
subject to
VlXl
bl, 2vxxl
-~- F2X2 ~
-
5112x2 ~ b2, ~ , ( v ~ ) / >
~, /~2(vtg) >t Gt,
with the membership functions O,
l%(vj) =
-- ~
< vj <~ v j1,
l _ { V- _s v2~ 2 \v11 - vj2 J '
vj <. vs <. v j,
1,
vj2 <<.vj <<.v j,3 j = 1,2,
1
(Vy-V~'~2 --
0,
\vj 4 - vs3 J
,
3 '
vj <~ v1 <.
2
• l~j ,
vj4 <~ vj < oo.
1. Assume t h a t b = ( 3 , 2 ) , c t = 0 . 2 , (vx, ~ vx, 2 vl, a vl) 4 = (0.2,0.5,0.7, 1.4) 2. The e-pareto optimal solution is (£, ~7) = (2.38,0.19,0.78,0.74) 3. The set of feasible parameters is:
U = {(b, vt)112.99 ~ 5bl + b2, 0.98 ~< 2bl - b2, 0.9v~ + 0.1v 3 >~ 0.9v~ + 0.1v 2 >/0.78, 0.9v24 - 0.1v23 ~> 0.9v~ + 0.1v 2 ~> 0.74}.
4. The solvability set. F r o m the strict convexity of x 2 + x 2, it follows that B = U. 5. The stability set of the first kind is G(£,~) = {1.99 ~< hi, 3.0098 ~< b2, 0.2 ,N
0.9v~ + O.lv~ i> 0.9vi + O.lv~/> 1.6, 0.9v~ + O.lv ] 1> 0.9v~ + 0.1v~ >1 8.3}. 6. The stability set of the second kind. Let us denote the stability set of the second kind of our problem corresponding to the index subset I ~ _ {1,2,3,4,5,6,7} by Q~ (i = 1,2 . . . . . 128), where 11 = {1}, /2 = {2},
M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343 351
351
I3 = {3},/4 = {4}, I5 = {5},/6 = {6}, I7 = {7}, I8 = {1,2}, I9 = {1,3}, 11o = {1,4},.... Then Q1 = {bi >~ 3, bE >/ --15, e = 1, 0.9v24 + 0.1v 3 >~ 0.9v~ + 0.1v 2 ~> 3}, Q2 --- {bl = 0, b2/> 0, e >~ 0, 0.9v 4 + 0.1v 3 ~> 0.9v2~ + 0.1v 2 ~> 4.5}, Q3 = {b~ ~> 0, b2 = 0, e >~ 0, 0.9v~ + 0.1v 3 ~> 0.9v~ + 0.1v 2 >~ 0}, Q, = {5bl + b2 ~> 0, e >~ 0.45, 0.9v~ + 0.1v 3 >~ 0.9v~ + 0.1VZl ~> 0.5}, Q5 = {5bl + bE >~ 0, e >/0.45, 0.9v~ + 0.1v 3 ~> 0.9v~ + 0.1v 2 >/0.7, 0.9v24 + 0.1v 3 ~> 0.9v2~ + 0.1v22 ~> 3}, Q6 = {5b~ + b2 ~> 0, e >~ 0.017, 0.9v 4 + 0.1v 3 t> 0.9v2~ + 0.1v 2 >~ 3}, Q7 = {5bl + bE /> 0, e >~ 0.25, 0.9v24 + 0.1v 3 ~> 0.9v~ + 0.1v 2 ~> 6}, Qa = {ba = 0, bE >~ 0, e -----0, 0.9v~ + 0.1v 3 /> 0.9v2~ + 0.1v 2 >/4.5}, Q9 = {bl /> 0, b2 = 0, e = 0, 0.9v 4 + 0.1v 3 ~> 0.9v~ + 0.1v~ >~ 0}, Qlo = {5bl + bE /> 0, e = 0.25,0.9v 4 + 0.1v 3 ~> 0.9v~ + 0.1v 2 ~> 0.5, 0.9v24 + 0.1v 3 /> 0.9v2~ + 0.1v 2 >~ 6}, similarly we can find the remaining sets of Q's.
8. Conclusion In this paper, we study the stability of multiobjective nonlinear programming problem with fuzzy parameters in the constraint functions which are characterized by fuzzy numbers, using the concept of ~-level sets of the considered fuzzy numbers. F o r the nonfuzzy problem we give the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind corresponding to a certain ~-level. Also, an algorithm for obtaining the stability set of the first kind corresponding to ct-pareto optimal solution is introduced. An investigation of the stability set of the second kind is presented.
References [1] M. Bazaraa, Nonlinear Programming (Wiley, New York, 1979). [2] V. Chankong and Y. Haimes, Multiobjective Decision Making: Theory and Methodology, North-Holland Series in System Science and Engineering, (Elsevier, New York, 1983). [3] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [4] O.L Mangasrian, Nonlinear Programming (McGraw-Hill, New York, 1969). [5] S. Orlovski, Multiobjcctive programming problems with fuzzy parameters, Control Cybernet 13(3) (1984) 175-183. [6] M.S. Osman, Qualitative analysis of basic notions in parametric convex programming I (parameters in the constraints), Appl. Math. CSSR Akad. Ved, Prague (1977). [7] M.S. Osman, A.Z.H. EI-Banna and I.A. Younes, On a general class of parametric convex programming problems, Adv. in Modelling and Simulation 5(1) (1986). [8] M.S. Osman and A.Z. El-Banna, Stability of multiobjective nonlinear programming problems with fuzzy parameters, Math. Comput. Simulation 35 (1993) 321-326. [9] R.T. Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math. 21 (1967) 167-181. [10] M. Sakawa and H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy Sets and Systems 29 0989) 315-326. I11] H. Tanaka and K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems 130) (1984) 1-10.
sets and systems Fuzzy Sets and Systems74 (1995) 343-351
ELSEVIER
Stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints M o h a m e d Abd E1-Hady Kassem*, Elsaid Ibraim A m m a r Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
ReceivedJanuary 1994;revisedJuly 1994
Abstract
This paper deals with the stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraint functions.These fuzzyparameters are characterized by fuzzynumbers. Qualitative and quantitative analysis of the basic notions like the set of feasible parameters, the solvability set, the stability sets of the first kind and of the second kind, will be reformulated under the concept of ct-pareto optimality. An illustrative example is given to clarify the obtained results. Keywords: Multiobjectivenonlinear programming; Stability;Parametric analysis;Fuzzy programming; Fuzzy numbers;
7-Pareto optimality
1. Introduction
Some basic notions like the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind for parametric convex nonlinear programming problems were introduced by Osman [6]. Tanaka and Asai [11] formulated multiobjective linear programming problems with fuzzy parameter. Orlovski [5] formulated a general multiobjective nonlinear programming problem with fuzzy parameters. Sakawa and Yano [10] introduced the concept of ~-pareto optimality of multiobjective nonlinear programming problem with fuzzy parameters. In this paper, qualitative and quantitative analysis of the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind for multiobjective nonlinear programming problem with fuzzy parameters in the constraints are studied by using the concept of at-pareto optimality. Also an algorithm for obtaining the stability set of the first kind which corresponds to ~-pareto optimal solution is introduced.
*Corresponding author. 0165-0114/95/$09.50 © 1995- ElsevierScienceB.V.All rights reserved SSDI 0165-01 14(94)00344-0
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2. Problem formulation Let us consider the following multiobjective nonlinear programming problem with fuzzy parameters in the constraints
P~,b:
minimize F(x) subject to x ~ X(b, ~) = {x ~ ~" I G(x, ~) <<.b},
where F : ~ " ~ R m, G : ~ " ~ R " are convex functions of class C (1) on ~", F = ( f l , f 2 . . . . . f~)T, G = (gx, g2, ".', g,)T, (bl, b2 .... , b,) T represent the vector of real numbers and ~ = (vl, v2,--., ~7~)represent a vector of fuzzy parameter in the constraints gj(x,~j), j = 1,2,...,r. These fuzzy parameters are assumed to be characterized by fuzzy numbers. The fuzzy numbers ~ form a convex continuous fuzzy subset of the real line whose membership function i~(v t) (t = 1,2, 3,4) is defined by [101: 1. a continuous mapping from ~ to the closed interval [0, 1]; 2. /~(v) = 0 for all v e ( - 0% vl]; 3. strict increase on Iv ~, v21; 4. /t~(v) = 1 for all v ~ Iv 2, v3]; 5. strict decrease on Iv 3, v*]; 6. /~(v) = 0 for all v ~ [v 4, ~). Here, we assume that the membership function/~(v) is differentiable on Iv x, v*] . Remark 1. It must be noted that if/~(v) is a concave function on [v 1, v*], the conditions (1)-(6) which are imposed on /~(v) are still satisfied and the membership function/J~(v) is a function of vt, t = 1, 2, 3, 4 at a certain level a. Definition 1. The a-level set of the fuzzy numbers ~j is defined as the ordinary set L~(q) for which the degree of their membership functions exceeds the level a: L~(~) = {vlp~j(vj) >~ a, j = 1, 2 . . . . . r}. For a certain degree a, the problem P~.b can be written in the following nonfuzzy form 1-91: aM:
minimize
F(x)
subject to
N(b, v') = {(x, v) E R"+rlgj(x, vj) ~ bj, j = 1, 2 , . . . , r; vj~ L~(~j), j = 1, 2 .... , r},
In the whole paper, we assume that the problem aM is stable [9]. Definition 2. x* ~ X(b, ~) is said to be an a-pareto optimal solution to the problem aM, if and only if there does not exist another x ~ X (b, ~), v ~ L~(q) such that F (x) ~ F(x* ), and F(x) ~ F(x*), where the corresponding values of parameters v* are called a-level optimal parameters. We see that the a-pareto optimal solutions of the problem aM can be characterized in terms of the a-optimal solution of the kth-objective e-constraint problem aM, [21, which takes the following form: aM~:
minimize
fk(X)
subject to
S,(b, ¢) = {(x, v) e R"+r IOj(X,vi) <<.b~, j = 1, 2 , . . . , r, vj ~ L~(~j), j = 1, 2,..., r, f~(x)~
where e = (el,e2, ... , e k - l , e k + l , ... ,era) T.
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345
Theorem 1. x* is an a-pareto optimal solution of problem aM if and only if x* solves ~M~ for every k =
1,...,m. Proof. Necessity. Assume that (x*, v*) does not solve aM,. for some k, then there exists (x, v) ~ NAb, v t) such thatfk(x) < fk(X*) implies that (x*, v*)~ N* (N* is the set of all a-pareto optimal solutions of the problem aM). Thus the conclusion follows. Sufficiency. Since (x*, v*) solves the problem aM,. for e = e* and every k = 1, 2 .... , m, then there is no other (x, v) ~ NMb, v t) such that f a x ) <~f~(x*), s = 1, 2 , . . . , m with strict inequality holding for at least one s. This implies by definition that (x*, v*) e N*. []
3. The set of feasible parameters Definition 3. (I) The set of feasible parameters for problem aM is denoted by U and defined by
U = {(b,v t) ~ ~5"[N(b,v') ~ 0}, (II) The set of feasible parameters for the problem aM~ is denoted by U~ and defined by U,: = {(e,b,v') e ~m+5r-'lN~(b,v') ~ 0}. Theorem 2. The sets U and U~ are convex.
Proof. T o prove the set U is convex, assume that (b 1, vtl), (b 2, V/2) are two points in U (v a = (v ~, ~)2i,v3i V4i), • llltl i = 1,2), then there exist the a-pareto optimal solutions x 1 and x 2 such that gj(x~,v~ 1) <<.b~, I~j~ j ) >>,a and ~j(X 2 ' yr.2 2 #~(v~2)/> a, j -- 1, 2 . . . . . k. Then for all 0 ~ 2 ~< 1 we have (1 - 2) Oj(x ~, v~~) + 2Oj(x 2, v~2) ~< j ) ~< b~, (1 - 2)b) + Abe, and (1 - 2")p~j~"vjt~") + 2 /%~,~.t2, ~ ~ >/(1 - 2)a + 2a = a. Since gj are convex functions for each j in x and/~:(v:) are concave membership functions in v, then for all 0 ~< 2 ~< 1 we have gj[(1 - 2)(x ~, v~x) + 2(x2,v~2)] ~< (1 - 2)b~ + 2b], j = 1,2 . . . . . r; and p~((1 - 2)v~ ~ + 2v~2)/> a, j = 1,2 ..... r. Then N[(1 - 2)b ~ +2b 2,(1-2)v '~+2v a]~0, i.e. [ ( 1 - 2 ) b x + 2 b z , ( 1 - 2 ) v " + 2 v ' 2 ] e U for all 0~<2~<1. Hence U is convex set. Similarly, we can prove that U~ is convex set. [] Theorem 3. I f gj(x, vj) and F(x) are lower semicontinuous for all (x, v), and t%(vj) are upper semicontinuous for
all v, then the sets U and U, are closed. Proof. Let us consider the point-to-set mapping F: U ~ R 5r which assigns a subset (b, v') ~ F(b, v ~) to each element (b, v') e U. Consider the sequences (b t"), v "")) e U, and (x t"), v "")) ~ R n+4" such that (x ~"),v "")) e F(b ~"~, v"")), gj(x ("), v~(")) <<.b~"), #~j(V~(n)) ~ ~ and b (") ~ b t°), (x ("), v "")) ~ (x (°), v"°)), v "") ~ v "°) as n -+ oo. F r o m the assumption, we have gj(x ~°), v~(°)) <~ limn-~ ® ~J, ,,.tx ~), v~t~)) ~< ~i, ho ~. j ~~.,(o)~ J J = lim,_,o~ p~j(v~t~)) >~ a. Then (x t°), v t°)) N(b ~°), v(°)), i.e., N(b (°), v ~°)) ~ O. Hence (b t°), v t°)) ~ U. Therefore U is closed. Similarly, we can prove that U, is closed. [] 4. The solvability set Definition 4. (I) The solvability set of problem a M is denoted by B and defined by
B = {(b,v') e Ulm,_opt(b,v')50}, where m~_op,(b, v') = {(if, 9)~ ~"+'13 a-optimal solution to the problem aM}
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346
(II) The solvability set of problem ~M~ is denoted by B, and defined by B~ = { (e,b, v') • U~l m~.ovt(e,b, v') ~ 0 } ,
where m~.opt(e,b,v') = {(x,v)~ ~"+rl q ~-optimal solution to the problem ctM,}. Theorem 4. (I) I f for one point (b, v t) ~ B such that the set m~_opt(b,v t) is bounded, then B = U. (II) I f for one point (e, b, v ~) ~ B, such that the set m~_opt(e,b, v ~) is bounded, then B, = U~. Proof. (I) From the assumption and [6], we have the sets A1 = {(e, b, vt) 6 Rm+4r-11N(e, b, v t) ~ 0}, and
A2 = { (e, b, vt) • A11 problem ~M~ has ~-pareto optimal solution} are equal. Since the projection of A~ and A2 in 5r-space are equivalent, B = U. Similarly, we can prove (II). [] Lemma 1. Under the assumption of Theorem 4, the sets B and B~ are convex.
The proof follows from Theorems 4 and 2. Lemma 2. I f the set B v~ 0 (B~ vL O) and either F (x) ( fk(X) ) is strictly convex on ~n or the set N (b, v') (N~(b, v') ) is bounded for one (b,v t) • U ((e,b,v t) • U,), then B = U (B~ = U,).
The proof follows directly from Theorem 4. Theorem 5. (I) The problem ~M has an ot-pareto optimal solution Y~ if and only if F(x) has a subgradient (b,v t) at (~,~), (b, v t ) • B such that b(x-~)>>.O, b(v-f)>>.O, v t ( x - ~ ) > ~ O , and v t ( v - f ) > ~ O for all (x, v) • N(b, vt). (II) The problem ~M, has an ~-optimal solution (~, ~) if and only if fk(X) has a subgradient (e, b, vt) at (,2, ~), (e, b, v t) e B, such that e(x - ~) >~ O, e(v - f) >1 O, b(x - Y~) >>,O, b(v - ~) >>.O, vt(x - ~) >1 0 and v'(v - ~) >_,0 for all (x, v) • N~(b, v').
The proof is similar to the one in [6].
5. The stability set of the first kind Definition 5. (I) Suppose that (b, v t) • B with a corresponding ~t-pareto optimal solution ~ and the or-level
optimal parameter q of the problem ~M. Then the stability set of the first kind corresponding to (2, q) which is denoted by G(~, f), is defined by G(~, ~) = {(b, vt) • B I(~, ~) is an ~t-pareto optimal solution of the problem ctM}. (II) Suppose that (g, b, qt) • B, with a corresponding ~-optimal solution (~, q) of the problem ~M,, then the stability set of the first kind for the problem ctM, is Q(~, q) = {(e, b, vt) ~ B, I(x, q) is an 0t-optimal solution of the problem ~M,}. Lemma 3. I f the set M is defined as
M = {(b, v') • B[gj(~, ~ ) <%bj, I%(~)/> ~, J = 1, 2..... r; t = 1, 2, 3, 4}, then it is equal to the set ~1 where
A4 - {(b, v t) e B] 3 e such that minfk(x) ~
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Proof. It follows from the fact that (Y, ~) e N(b, v'), where ~ is an ~-pareto optimal solution and ~ is the or-level optimal parameter of the problem ~M, for every (b, v') • M (and from [5]) that the set
= {(e,b,v') • B~[fs(Y) ~< es, s = 1,2 . . . . . m, s # k, Oj(ff, 9~) ~< bj,/~j(v~) >/~,j = 1,2, ... ,r;t = 1,2,3,4}, and the set
]fl = {(e,b,v')• B~,
min
(x, v)eN,~(b,vt)
fk(X' <~fk(X)}
are equal. Since the projections of the above sets A~t in 5r-space are equivalent, the result follows.
[]
R e m a r k 2. F r o m L e m m a 3, it follows that G(~, 9) _ M. T h e o r e m 6. The sets G(F,,9) and G~(F,,~) are star shaped with common point of visibility b i = 9j(~,gj),
= #~j(9~), j = 1, 2,..., r. Proof. Since (b, ~') is c o m m o n point of visibility of G(~, ~), i.e., Oj(~, fj) = bj, /z~j(9~) = ~, j = 1, 2 . . . . . r;
t = 1,2,3,4. Assume that (b°,v °') • G(~,9), then there exists an index set J _ {1,2,...,r} such that gj(~, 9~) = b °, #~j("vj°'') = a, j • Jo; gj(x, 9j) <~ b °, # ~jvj ( .o,,j > ~ & j • { 1 , 2 , . . . , r } - J o . F r o m the convexity of g)(x, vj) and concavity of/%(vj) we get for0~<2~
9j[(l-2)x
1+2x z,(1-2)v)
+ 2 v 2]
~< (1 - ~.)gi(xl,vl) + 2gj(x2, v]),
(1)
zvt2x #~j[(1 - 2)v~ ~ + 2v~2 ] / > (1 - 2)/~,(v~ ~) + /~/a~,~ j j.
(2)
F r o m the assumption, we have the following two equations for j • Jo; (1 -- 2)9j(~, ~ ) + ~gj(x,- ovy) -- (1 - 2)b~ + 2b ° = /gj, (1 - 2 ) # ~ ( ~ ) + 2#~(v °') = (1 -- 2)~ + 2~ = a,
for j • Jo,
j • So;
(3) (4)
and the following two equations for JCJo: (1 - 2)gj(ff, f~) + 29~(g,v °) <-%(1 - 2)bj + 2b ° = b~, for jCJo,
(5)
(1 - 2)/~(f~) + 2#~(v °') ~> (1 - 2)~ + ~.0~= ~,
(6)
for jCJo.
F r o m (1), (3), (5) we get
gj[2x' -t- (1 - 2)x z, 2v~ + (1 - 2)v~] = O)(~,vS) = b), j • Jo; gj[2x a + (1 - 2)x z, 2v) + (1 - 2)v~] = gj(~, f)) ~< b), j • Jo, where :~ = 2x ~ + (1 - 2 ) x z, f ) = 2 v ) ' + (1 - 2 ) v ~ ' . Also from the equations (2), (4), (6) we get / ~ ( ~ j ) = ~, for j • Jo, /~)(f~)/> ~, for JCJo, where ~' = 2f' + (1 - 2)v °'. Then the point (b, ¢') = [(1 - 2)b + 2b °, (1 - 2)9 t + 2v *t] • G(ff,,7). Hence the result. [] Similarly, we can prove that the set G,(~, ~) is star shaped.
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348
6. Determination of the stability set of the first kind
Let (b, qt) • B with an a-pareto optimal solution £, the corresponding Karush-Kuhn-Tucker conditions will then have the following form:
~(~, ~j) ~fk(£) (~f~(2) Ox---~-+ ~ v , ~ - x p + L uj - - - 0 , i =s
j=1
(~x~
fl = 1, 2,...,n,
(7)
.....
(8)
k#s
~g~(~,~j)
3
~(~j)
~V0
j=1
~Y0
j= 1
f,(£)~
r,
s = l , 2 ..... m, s # k ,
(9)
gj(£, ~j) <~ bj, j = 1, 2 ..... r,
(10)
/%(~j) t> ct, j = 1,2,...,r,
(11)
vs[fs(£) - e~] = 0,
(12)
Uj[gj(X, f j ) --
bj]
s = 1,2 .... ,m; s # k,
=0,
j = 1,2 ..... r,
(13)
6j[#~,(~Tj) - ~] = 0, j = 1,2 ..... r,
(14)
v~>0,
(15)
s = l , 2 ..... m, s v~ k,
uj >~ O, 6j >~ O,
j = 1 , 2 ..... r,
(16)
From the Eqs. (7), (8), (15) and (16), we can determine the value of uj and 6j. According to these values, the stability set of the first kind G(£, q) will be determined as follows: (1) I f u j = O , J j = O , j = l , 2 ..... r, GI(£, f) = {(b,v')lgj(£, ~j) ~< bj, t%(~})/> a}(2) I f u j > O , 6 j > O , j =
1,2 ..... r,
Gz(£,q) = { ( b , v ' ) l g j ( £ , f j ) = bj, t%(f}) = ~}. (3) I f u j > O , 6 j = O , j = l , 2
bj>0, j=
(18)
..... r,
Ga(£, ~) = {(b, v')lgj(£, 5 ) = bj, #~(q))/> a}. (4) I f u j = 0 ,
(17)
(19)
1,2, .... r,
64(£, f) = {(b, vt)lg~(£, qj) <~ bj; #~(ft) = a}.
(20)
(5) If uj = O, t~j = O, j • J ~ {1,2 ..... r}, and uj > O, t~j > O, jq~J, G~(£, f) = {(b, v')I gj(£, fj) <~ b j , / % ( ~ ) >/a; and gj(£, ~j) = bj,/~(q~) = a}.
(21)
Gs(£,q).=
(22)
Thus U
a,(£,~).
possible J
The algorithm
(1) For a certain a • [0, 1], we select an arbitrary (b,~')• B and solve the problem aM to obtain an :t-pareto optimal solution ~. (2) Formulate the K a r u s h - K u h n - T u c k e r conditions (1)-(10).
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349
(3) From the Eqs. (7), (8), (15) and (18), we determine the values of u~ and 6j by using a suitable algorithm. (4) According to the values of u~ and 6~, the stability set of the first kind G(~, f) can be determined as in the Eqs. (17)-(22).
7. The stability set of the second kind Definition 6. (I) We suppose that (b, ~t) ~ B with a corresponding a-pareto optimal solution ~ of the problem aM such that (£,~) ~ a(b, qt;J), where
a(b, q'; J) = {(x, v) ~ E"+r Lgj(x, vj) = [~j, I~j(q)) = a, j ~ J c {1, 2..... r} and
gj(x, vj) <~ bj, la~j(9)) >~ a, jCJ}. Then, the stability set of the second kind corresponding to a(b, 9t; j ) denoted by Q(a(b, ~t; j)) is defined by
Q(a(b,~';J)) = {(b, v') e BIE(b,v')c~a(b,9';J)
# 0},
where E(b, v') denotes the set of all a-pareto optimal solutions of problem aM. (II) We suppose that (g,b,9')~B~ with a corresponding a-optimal solution Y of ~M~ where (y, ~7)~ a~(b, 9'; J), where a~(b,9'; J ) = {(x,v) ~ R"+r I f~(x)= g~, S ~ I = {1,2, ... ,m}, s ¢ k,
gj(x, vj) = [gj, p~j(q)) = a, j 6 J = {1,2,...,r} and L(x) <~ g~, s¢l, s ~ k, gj(x, vj) <~ bj, #~j(9)) >1 a, jCJ}. Then, the stability set of the second kind corresponding to try(b, 9'; J) denoted by Q(ae(b, ~'; J)) is defined by
O~(ae(b, 9'; J)) = {(~, b, v') ~ B~ I E(~, b, v') c~ ae(b, 9'; J) 4: 0}, where E(e, b, v t) denotes the set of all 0t-optimal solutions of problem aM~. Remark 3. From the definitions of the stability sets of the first kind and of the second kind, we have (I) Q(a({9, 9'; J ) ) = U ~* G( xi, v~), where I = { i l (x i, v i) ~ a(b, ~'; J) and x i is an a-pareto optimal solution of the problem aM}. (II) Q~(a~([~,~';J))= Ui~1G(x~,vi), where I = {i[(xi, vi)~ae(b, gt;J) and x i is an a-optimal solution of problem the aM~}. Theorem 7. (I) I f the functions J~(x), i = 1,2, ...,m are strictly convex on ~" and (b 1, ¢1), (b2, v,2) are two
distinct points in B with Q(a(bl, vtl; J~)) 4: Q(a(b2,vt2; J2)). Then Q(a(b 1, vt~; J1 )) 0 Q(a(b 2, v '2, J2)) :/: 0. (II) If the function fk(x) is strictly convex on ~" and (e, b ~, v'~), (e, b 2, v t2) are two distinct points in B~ with Q,(a~(ba,vt~;J~)) ve Q~(a~(b2, v'2;J2)). Then
Q~(a~(b~, vtl; J~))~Q~(a,(b2, vt2; J2)) ¢ O. Proof. 0) By the assumption, it follows that J ~ J 2 . Suppose that Q(a(b 2, v'2; J2)) then there exists an a-pareto optimal solution if, such that (Y, 9) e tr(b, "7';J~ ) & a(b, ~'; Ja),
([~,f*)~Q(a(b~,vta;J~))~
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M.A.E. Kassem, E.I. A m m a r / Fuzzy Sets and Systems 74 (1995) 3 4 3 - 3 5 1
where
a(b,~';J1) = {(x,v) e R"+'lo~(x,v~) = b~, #~j(j) ~' = ~ , J ~ J 1 ; g j ( x , v j ) < D j , l ~ j ( ¢ J ) > ~ , J C J a } , tr([~, U; Jz) = {(x, v) ~ ~"+" [gj(x, vj) = b j, #~s(~) = ~, j 6 J2; gj(x, vj) < bj, p~j(~}) > 0~,JqsJ2 }. Therefore, 9s(x,v~) = b~, l~o,(g~) = ~; 9~(x, vs) < b~,/~s(g~) > ~, for at least one s • {1,2 . . . . . r} if J1 ¢ Jz, and then J1 = Jz which is contradiction. Hence the result follows. Similarly, we can prove (II). [] Example. Consider the following fuzzy problem: P{~.b):
minimize
(X 2 -- X2, X 2 At- X 2 )
subject to
~lX~ + ~2x2 ~< bl, 2~71x~ - 5~2x2 ~< b2.
The equivalent nonfuzzy problem takes the form: eM:
minimize
(x 2 - x 2 , x 2 + x 2)
subject to
VlXl
bl, 2vxxl
-~- F2X2 ~
-
5112x2 ~ b2, ~ , ( v ~ ) / >
~, /~2(vtg) >t Gt,
with the membership functions O,
l%(vj) =
-- ~
< vj <~ v j1,
l _ { V- _s v2~ 2 \v11 - vj2 J '
vj <. vs <. v j,
1,
vj2 <<.vj <<.v j,3 j = 1,2,
1
(Vy-V~'~2 --
0,
\vj 4 - vs3 J
,
3 '
vj <~ v1 <.
2
• l~j ,
vj4 <~ vj < oo.
1. Assume t h a t b = ( 3 , 2 ) , c t = 0 . 2 , (vx, ~ vx, 2 vl, a vl) 4 = (0.2,0.5,0.7, 1.4) 2. The e-pareto optimal solution is (£, ~7) = (2.38,0.19,0.78,0.74) 3. The set of feasible parameters is:
U = {(b, vt)112.99 ~ 5bl + b2, 0.98 ~< 2bl - b2, 0.9v~ + 0.1v 3 >~ 0.9v~ + 0.1v 2 >/0.78, 0.9v24 - 0.1v23 ~> 0.9v~ + 0.1v 2 ~> 0.74}.
4. The solvability set. F r o m the strict convexity of x 2 + x 2, it follows that B = U. 5. The stability set of the first kind is G(£,~) = {1.99 ~< hi, 3.0098 ~< b2, 0.2 ,N
0.9v~ + O.lv~ i> 0.9vi + O.lv~/> 1.6, 0.9v~ + O.lv ] 1> 0.9v~ + 0.1v~ >1 8.3}. 6. The stability set of the second kind. Let us denote the stability set of the second kind of our problem corresponding to the index subset I ~ _ {1,2,3,4,5,6,7} by Q~ (i = 1,2 . . . . . 128), where 11 = {1}, /2 = {2},
M.A.E. Kassem, E.L Ammar / Fuzzy Sets and Systems 74 (1995) 343 351
351
I3 = {3},/4 = {4}, I5 = {5},/6 = {6}, I7 = {7}, I8 = {1,2}, I9 = {1,3}, 11o = {1,4},.... Then Q1 = {bi >~ 3, bE >/ --15, e = 1, 0.9v24 + 0.1v 3 >~ 0.9v~ + 0.1v 2 ~> 3}, Q2 --- {bl = 0, b2/> 0, e >~ 0, 0.9v 4 + 0.1v 3 ~> 0.9v2~ + 0.1v 2 ~> 4.5}, Q3 = {b~ ~> 0, b2 = 0, e >~ 0, 0.9v~ + 0.1v 3 ~> 0.9v~ + 0.1v 2 >~ 0}, Q, = {5bl + b2 ~> 0, e >~ 0.45, 0.9v~ + 0.1v 3 >~ 0.9v~ + 0.1VZl ~> 0.5}, Q5 = {5bl + bE >~ 0, e >/0.45, 0.9v~ + 0.1v 3 ~> 0.9v~ + 0.1v 2 >/0.7, 0.9v24 + 0.1v 3 ~> 0.9v2~ + 0.1v22 ~> 3}, Q6 = {5b~ + b2 ~> 0, e >~ 0.017, 0.9v 4 + 0.1v 3 t> 0.9v2~ + 0.1v 2 >~ 3}, Q7 = {5bl + bE /> 0, e >~ 0.25, 0.9v24 + 0.1v 3 ~> 0.9v~ + 0.1v 2 ~> 6}, Qa = {ba = 0, bE >~ 0, e -----0, 0.9v~ + 0.1v 3 /> 0.9v2~ + 0.1v 2 >/4.5}, Q9 = {bl /> 0, b2 = 0, e = 0, 0.9v 4 + 0.1v 3 ~> 0.9v~ + 0.1v~ >~ 0}, Qlo = {5bl + bE /> 0, e = 0.25,0.9v 4 + 0.1v 3 ~> 0.9v~ + 0.1v 2 ~> 0.5, 0.9v24 + 0.1v 3 /> 0.9v2~ + 0.1v 2 >~ 6}, similarly we can find the remaining sets of Q's.
8. Conclusion In this paper, we study the stability of multiobjective nonlinear programming problem with fuzzy parameters in the constraint functions which are characterized by fuzzy numbers, using the concept of ~-level sets of the considered fuzzy numbers. F o r the nonfuzzy problem we give the set of feasible parameters, the solvability set, the stability set of the first kind and the stability set of the second kind corresponding to a certain ~-level. Also, an algorithm for obtaining the stability set of the first kind corresponding to ct-pareto optimal solution is introduced. An investigation of the stability set of the second kind is presented.
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