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On some basic notions of fuzzy parametric nonsmooth multiobjective nonlinear fractional programming problems
sets and systems ELSEVIER
Fuzzy Sets and Systems99 (1998) 291-301
On some basic notions of fuzzy parametric nonsmooth multiobjective nonlinear fractional programming problems Elsaid Ebrahim Ammar Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
Received March 1996;revisedJanuary 1997
Abstract
Different basic notions like the solvability set, the stability set of the first and of the second kind for the smooth fuzzy parametric multiobjective NLP programs have been discussed in several papers. In this paper, these notions have been defined and qualitatively analyzed for a general class of nonsmooth multiobjective nonlinear fractional programming problems with fuzzy parameters under the concept of ct-pareto optimality. The stability sets of the third and of the fourth kinds have also been defined and analyzed for this problem. A determination of the stability sets of the third and of the fourth kinds will be given. © 1998 Elsevier Science B.V. All rights reserved Keywords: Multiobjective nonlinear fractional programming; Nonsmooth functions; Parametric approach; Fuzzy para-
meters; ~t-pareto optimality; Stability
1. Introduction
Application and algorithms for fractional programming have been treated in considerable detail since the early work of Isbell and Marlow 1,9]. Among the many applications are the portfolio selection, stock cutting, game theory and numerous decision problems in management science. See Grundspan r5], Schaible 1,18], and Pardalos and Philips [16] for the most recent surveys. In earlier work, Osman 1,13] introduced the notions of the solvability set, stability set of the first kind and stability set of the second kind, and analyzed these concepts for parametric convex nonlinear programming problems. Recently, Tanaka and Asai [19] formulated multiobjective linear programming problems with fuzzy parameters. Orlovski [12] formulated general multiobjective parameters. Sakawa and Yano [16] introduced the concept of ~-pareto optimality of fuzzy parametric programs. In [1] we studied a multiobjective nonlinear programming problem with fuzzy parameter in the constraints functions. Also Osman and E 1Banna [14] introduced an algorithm for obtaining the subset of the parametric space which has the same corresponding ct-pareto optimal solution. In this paper, a qualitative analysis of the notions the solvability set, the stability sets of the first kind and of the second kind of the multiobjective nonlinear fractional programming problems (FMFP) with fuzzy parameters in the ratio of the objective functions without differentiability assumptions are presented. 0165-0114/98/$19.00 © 1998 ElsevierScienceB.V. All rights reserved PII S0165-0114(97)00033-X
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A parametric analysis of the equivalent nonfuzzy (eMFP) problem (for a level-s) of this problem is given. A determination of the stability sets of the third and of the fourth kinds of e M F P problem will be introduced.
2. Problem formulation
Consider the following fuzzy parametric multiobjective nonlinear fractional programming problems: FMFP(~i,b)
min
Fl(x, 41,bl) - f x ( x ' 4 1 ) gl(x, bx) F2(x ' 42 , t~2) __ f2(X, 42) g2(x, bE)
Fk(x, ak, bk) - A(x, a,) gk(x, b,)' s.t.
x~M={x~R"lhj(x)<~O,j=l,
2 . . . . . m},
wherefi(x, ai) and gi(.~, bi), i = 1, 2, ..., k are nonsmooth convex and concave functions on E"+k,f/(x, 4i) >~ 0 and gi(x, bi) > 0 for each i = 1, 2. . . . . k, respectively. ~ii = (41,42, ... ,ak), /~i = (bl, b2. . . . . bk) represents the vectors of fuzzy parameters in the objectivesj~(x, 4~, bk), i = 1, 2. . . . . k. These fuzzy parameters are assumed to be characterized as the fuzzy numbers introduced in [4]. The functions hi(x), j = 1. . . . , m are nonsmooth convex functions on ~". Definition 1 (Sakawa and Yano [17]). The s-level set of the fuzzy numbers 41, bl are defined as the ordinary set S~(ti, b) for which the degree of their membership functions exceeds the level e: S~(&/~) = {(a,b) e 1~2kll~,(ai) >>.e, #b,(bi) >~ e, i = 1,2 . . . . . k}. For a certain degree of e, the problem (FMFP) can be written in the following nonfuzzy form [16]: eMFP(a, b) min s.t.
(Fl(x, al, bl), F2(x, a2, b2) . . . . . Fk(X, ak, bk)) T X e M and (a, b) e S,(4, b).
In this paper, we assume that the problem eMFP(a, b) is stable [2, 14]. Definition 2. A point ~(a, b) ~ M is said to be an e-pareto optimal solution of the eMFP(a, b) problem if and only if for at least one (d, b) s S~(&/~), there does not exist another x(a, b) ~ M such that
F,(x, d,, b,) - fi(x, d~) <~fi(2, d~) _ F,(2, d,, bi), gi(x, bi) gi(x, bi)
i = 1, 2, ..., k
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293
with strict inequality for at least one i e {1, 2, ..., k } holds. In [3] Dinkelback shows that the solution of problem aMFP(a, b) is a solution of the following auxiliary problem ctMFP(a, b, c): ~tMFP(a, b, c)
min
Z~(x, a~, b~, c~) =f~(x, a~) - c~g~(x, b~),
min
Z~(x, a2, b2, c2) =f2(x,
min
Zk(X, ak, bk, Ck) = fk(X, ak) -- Ckgk(X, bk),
s.t.
X e M, (a, b) ~ S~(t~,b),
a2) --
C2g2(X, b2),
where c~ ~ ~k, i = 1, 2, ..., k are nonnegative real numbers. The range [_ci,?i] of ci, i = 1, 2 . . . . , k can be determined from the set of pay-off value of the problem ~tMFP(a, b, c) as in I-8]. Definition 3. A point ~(a, b, c) ~ M is said to be an ~-pareto optimal solution of ctMFP(a, b, c) if and only if for at least one (~i~, b~) ~ S,(ti, b), ~ ~ ~k and ~ >/0, i = 1, 2, ..., k there does not exist another x(a, b, c) ~ M such that
Z~(x, ~, b~, ~) ~< Z~(~, 81, b~, ~)
for i = 1, 2, ..., k
with strict inequality for at least one i ~ { 1, 2 . . . . . k) holds. The above problem ~MFP(a, b, c) can also be reformulated in the weighting problem P(a, b, c, co) as in the following form: k
P(a, b, c, co) rain
~ coiZi(x, ai, bi, ci) i=1
s.t.
x ~ M,
(a, b) ~ S,(8, b),
where O = {~i/> 0, i = 1, 2. . . . , k, coi ~ 0, E~= ~ co~ = 1}. Definition 4. A point ~(a, b, c, co) ~ M is said to be an optimal solution of the P(a, b, c, co) problem if and only if for at least one (ti, b) ~ S,(8, b), 3i ~ ~k, e3i >/ 0, i = 1, 2. . . . ,k, and o3i :~ 0, we have k
k
i=1
i=1
It is well known from [2, 7] that the ct-optimal solution x(2, #, v, co) of the weighting problem P(a, b, c, co) is an 0t-pareto optimal solution of~MFP(a, b, c) problem if and only if at least one of the following conditions are valid: (1) Problem P(a, b, c, co) has a unique solution. (2) All the weights of the weighting problem concerned are strictly positive. Let us consider the following sets: E(a, b) = {~ ~ •" 1~ is ~-pareto optimal solution of 0tMFP(a, b)}, E(a, b, c) = {~ e Rnl~ is ~t-pareto optimal solution of ~MFP(a, b, c)}, and E(a, b, c, co) = {x e ~nlx is an optimal solution of P(a, b, c, co)}, are the set of ~-pareto optimal solutions of the ~MFP(a, b), ~MFP(a, b, c) and the set of or-optimal solutions of P(a, b, c, co), respectively.
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3. The solvability set
Definition 5. The solvability set B of problem aMFP(a, b) is defined by B = {(a, b) • R2kl Problem (aMFP) has the a-pareto optimal solution}. Remark 1. B = {(a, b) • R2k[ E(a, b) ~ O}. Lemma 1. If E(a, b, c, o9) is a bounded set, then B is a convex set. Proof. Suppose that (~V, b ~) • B, e = 1, 2 with a corresponding to a-level of the problem aMFP(a, b), then there exist 5 , , i = 1 , 2 . . . . . k are nonnegative real numbers with Z~(~,de, be, d,)=O and o3,~>0, i = 1, 2 . . . . . k, 03~ ~ 0 such that
(-SiZi(x, al, b~, el) <~ ~ (-SiZi(x, a~, b:, ci), i=1
i=1
and
~, cfiZ,(~, a 2, b 2, 5,) <~ ~, (SiZi(x, a 2, b 2, ci), i=1
i=1
for all x(a, b) • M, (a, b) • S~(5, b). Therefore, for 0 ~< 2 ~< 1 and by the convexity of Z,, i = 1, 2 . . . . , k, we have
~iZ,(~, dti, bi, ~) <~ i=1
(SZi(x, di, bi, 5,), ,=1
for all x(tl, 6) • M, where ai = (1 - 2) a/1 + 2a 2 and/~ = (l - 2)b I + 2b 2, then (~,,/~,) • B, and therefore B is a convex set. []
4. The stability set of the first kind
Definition 6. Given a certain (d, b) with a corresponding a-pareto optimal solution ~ of the problem aMFP(a, b), then the stability set of the first kind $1 (~, d, b) of aMFP(a, b) corresponding to ~ is defined by Sl(g, 5, b) = {(a, b ) • R2kl~ is an a-pareto optimal solution of problem aMFP(a, b)}. Remark 2. The stability set of the first kind of the aMFP(a, b) problem is the set of all parameters corresponding to one a-pareto optimal solution. Theorem 1. The set $1 (~, ti, b) is convex.
Proof. If $1(~, d, b) is empty or a single-point set, then the proof is trivial. Suppose that (a 1, bl), (a 2, b 2) • $1(~, d, b), then there exist g, >~ 0 and aS, t> O, i = 1, 2, ..., k, o3, ¢ 0 such that k
Y. i=1 k
k
al ) -
bl ) ] <. Y.
)-
/=1 k
a3,[j](~, a 2) - 5,g,(~, b~)] ~< ~ 03i[A(x, a 2) - 6,g,(x, b2)], i=1
i=1
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for x e M, a n d Zi(~, a~, be, 8~) = 0, e = 1, 2, i = 1, 2, ..., k. F r o m the convexity ofJ~, g~ in a, b if 2 e [0, 1] a n d ~i = 2a/~ + (1 - 2)a 2, b, = 2b~ + (1 - 2)b~, we get k
k
a , Ef,(~, a,)
-
e,g,(~, bi)] ~< ~] o3,[A(x, at) - ¢,o,(x, ~,,)]
i=1
i=1
a n d Zi(~, ~i, bi, ~) = 0, i = 1, 2, ..., k, i.e. (~, b) e S~ (~, 8, b), a n d hence the result follows.
[]
Theorem 2. The set S~(~, ~, [~) is closed. Proof. Assume that the sequence (a t"), b ¢n~)~ S1(~, ti, b), tai " ~) , oi ,tn),~ ~ ~ai , (o), b~°)) as n -+ o~, then there exist ~, ~ ~*, ~ t> 0 a n d O) i ~ O, i = 1, 2, ..., k, e31 4: O, such that k
k
e3i[f/(~, al n)) -- ~,gi(~, bln))] ~< ~ o3[f~(x, al n)) - ~ig,(x, bln))] i=1
i=1
a n d Zi(x,- a~tn) , ,in) - , = 0. By taking the limit as n --, oo, we get o~ , co k
o3,[f~(.~, ai(o)~, - -
k
8ig,(#,
b]°))] ~< ~
i=l
and
~,[f~(x,
al °)) - c',#i(x, bl°))]
i=1
Z / ' . ~ ~(0) L(O) - x
i~ , -i , vi , co = O, then (a t°), b t°)) e S~(~, ti, b) a n d hence the result follows.
[]
T h e o r e m 3. I f Ft(x, ai, bi), i = 1, 2, ..., k are strictly convex functions on M, then
Sl(x, d, b)nSl(x*, a*, b*) = O, where f~, x* are two distinct o~-pareto optimal solutions of 0tMFP(a, b). Proof. Let Sl(~, a, b)c~Sl(x*, a*, b*) ~ O, then (a, b) e S l ( X , a , b ) ( " 3 S l ( x * , a*, b*), there exist m o r e t h a n one ~-pareto optimal solution for ~ M F P ( a , b), then we have the following two systems:
and
[J;(x, ai) - e~gi(x, b~)] - [A(x*, a~) - ~o~(x*, b,)] ~< 0 with strict inequality for at least one i, have no solution x(a, b)~ M, (a, b ) e S,(a,/~). Since Fi(x, a~, bO, i = 1, 2, ... ,k are strictly c o n v e x functions, then there eixst ~ >I 0 and r/~ >t 0, i -- 1, 2 . . . . ,k such that
~,[Y;(x, a,) - e,g,(x, b,)] >/~,[f,(~, a,) - ~,g,(~, b,)] and
ni[f~(x, ai) - e~gi(x, b~)] >/~i[f~(x*, ai) - e~o~(x*, b~)]. F o r 2 ~ [0, lJ, we have (1 - 2 ) ¢ ~ [ y ; ( x ,
ai) - ~ o i ( x , b~)l/> (1 - ,~00¢m,[J](~, a~) - e~o~(~, b~)]
>/(1
--
,,1.)~ir/i[J;(.,q, ai) -- cigi(.x, bi)]
- 2cigi(x, hi),
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E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
and
2~,r/,[f~(x, a~) - e,0,(x, b~)/> 2~,ni[jS(x*, a~) - eig~(x*, b~) >/,t~,tt,[jS(x*, 43 - e~o,(x*, b~)] - (1 -
,~)¢ig~(x*, [,,).
Consequently, (irh[f~(x, d~) - 2?~g~(x, b,)] ~> (it/~[f~().2 + (1 - 2)x*, di) -- 2~g~(2# + (1 -- 2)x*, bi)].
Putting k
k
then 03, > O, ~ 03, = 1
(,rh/~,, i = 1, 2, ..., k,
(,r/i = i=l
i=l
and k
k
03,Ef~(2~ + (1 - 2)x*, ~i~)- 2~g,(2ff + (1 - 2)x*, b,)],
03/Eli(x, gt,) - 2e, gi(x, b,)]/> i=1
i=l
which leads to a contradiction, hence the result follows.
[]
5. The stability set of the second kind Definition 7. Given a certain (ti, b) c B with a corresponding ~-pareto optimal solution ~ of the problem 0~MFP(a, b), I c {1, 2 . . . . ,k}, J c {1, 2 . . . . ,r} and ~ c 0(& b; (I, J)), where 69(4, b; (I, J)) = {(x, a, b) c R"+Zkll~n,(al) = ~, #b,(bi) = ~, i ~ I _ {1, 2 . . . . , k}, hi(x) = O,j ~ J c_ {1, 2. . . . . r} and pn,(a~) > ~, pb,(bi) > ~, i•I, hi(x) < O, jq~J},
then the stability set of the second kind S2(0(a, b; (I, J))) of the problem ~MFP(a, b) is defined by Sz(0(& b; (I, J))) = {(a, b) 6 B I 0(ti, b; (I, J)) contains an a-pareto optimal solutions of problem (aMFP)}. Remark 3. The stability set of the second kind of the problem ~MFP(a, b) represents the set of all parameters (a, b) c R 2k for which corresponding to each one of them there exist an a-pareto optimal point such that all this a-pareto optimal points either rest on one side of the restriction set M or it is interior, in other words, it is the set of all parameters corresponding to one side of the feasible region M if this side contains an a-pareto optimal solution of ~MFP(a, b). Remark 4. It follows from Definitions 6 and 7 that S2(0(d, b; (I, J))) = U $1(~', d, b), ,Eli
where I' = {i1 ~i ~ O(& b; (I, J)) is an ~-pareto optimal solution of aMFP(a, b) problem}.
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297
Remark 5. From Definition 6, it follows that
s2(o-(a, b; (I, J))) = [J s2(e(a, b; (I, J))), JEA
where A = { j [ J <<.Jj and 0-(5, b; (I, J)) is the closure of 0(a, b; (I, J)}. Theorem 4. l f Fi(x, ai, bi), i = 1, 2 . . . . . k are strictly convex function, 11 ~ Iz, and J1 ~ J2, then
$2(Q(4 , b; (11, J1))t"~S2(Q(4, b; (•2, J2)) : 0. Proof. Suppose that (a, b) c S2((~)(d, b; (11, J1))c~Sz(p(4, b; (12, J2)), then E(a, b, c, co)rip(4, b; (11, J1)) :~ 0 and E(a, b, c, co)n0(d, b; (12, J2)) # 0. Since E(a, b, c, co) is a single point and 11 # I2, Jx # J2 then the above two relation tend to a contradiction, hence the result follows. [] Lemma 2. Under the assumption of the above theorem, S2(0(ti, b; (I, J))) is disconnected. The proof follows from Theorem 4 and Remark 4. Lemma 3. I f I' is a finite set, then $2(0(4, b; (1, J)))u{0, 0} is a closed convex set. The proof follows from Theorem 2 and Remark 4. Remark 6. A set T is said to be star shaped if there exists a point £ c T such that for all x c T, the closed line segment [2, x] e T, and £ is said to be a point of common visibility. Theorem 5. The set $2(0(& b; (I, J)))w{0, 0} is star shaped, with the point (a, b) = (0, O) its common visibility point. Proof. Let (a, b) e $2(0(4, b; (I, J))), then from Remark 3, (a, b) e $1(£ i, 4, b) for at least one index i e 1'. Then for each ~ > 0, we get (~a, eb) = e(a, b) c S2(p(d, b; (I, J))), and hence the result follows. []
Remark 7. From Definition 7 it follows that
82(Q-(4 , b; (1, g))) : (J S2(Q(~ i, bi; (1, J))), i~A
where A = {(i,j)lI <~ Ii, J <~ Ji and Q-(fi, b; (I, J)) is the closure of Q(& b; (I, J)}. Theorem 6. Under the f r s t assumption of Theorem 5 and if $2 (0 (ti, b; (I, J))) = B w {0, 0}, then
S2(p(a, [~; (I, J))) = S:(O-(4, [~; (I, J)))~J{0, 0}, where $2 (p-(5, b; (I, J))) and $2 (P(a, b; (I, J))) are the boundary and the closure of S2(Q(5, b; (1, J))), respectively. Proof. If either $2(0(5, b; (I, J))) is closed or $2(Q(4, b; (1, J))) = $1(2, 4, b), then the result is clear. Let (4, b) be a boundary point of $2 (~)(5, b; (I, J))), if (a, b) = (0, 0), also the result is clear. Otherwise, choose a sequence
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
298
(a ¢"), b¢n)) >/(0, 0), which converge to (a <°), b ~°)) as n ~ oo then there exist, 6~/> 0, 05~ ~> 0, i = 1, 2, ..., k. 05~ ¢ 0 such that (a ~n),b ~"~)~ SE(~(a, b; (I, J))) with corresponding to ~-pareto optimal solution x ~") ~ 0(~i, b; (I, J)), then k
k
~. 05, Ej'~(x~n', al n))
-
(.,g,(x ~n),bl"))] ~< ~ 05,Eft(x, al n))
i=1
-
(~9,(x, bln))]
i=1
and Z~(x ~n),u~-~n),~,h!n), 6~) = 0 for all x ~ M and for all n. By taking the limit as n ~ oo and from the finiteness of the sum and the continuity o f f and 9~, it follows that 05, f i=1
lim x ("), al °) - ~g~ lim x ("~, bl °) \n'-* ~
n-~°°
~< ~ 05, [j~(x,
- ~g,(x, bl°))]
i=1
and Z~ (limn-. ~o x t"), al °), ~,h!°),ci) = 0 for all x(a, b) ~ M, i.e., k
k
o3, Ef/(x ~°), al °)) - (.,gi(x ~°), b]°))] ~< ~ co, Efi(x, a] °)) - (~igi(x, bin))] i=I
i=1
and Z~(xt°),al °), bl °), c~)= 0 where lim, o~ x t") = x t°) exist since Bw{0,0} is closed and it is a-pareto optimality by the fact that the E(a t°~, b t°), g, 05) = {xt°)}. Therefore x ~°) E o(a ~°), bt°); (I, J)) and this completes the proof. []
6. The stability set of the third kind
Definition 8. Given a certain (d, b) ~ B with a corresponding a-pareto optimal solution ~ of the problem aMFP(a, b), x*(a, b) ~ M is any feasible point for a level-a, and 6 > 0; then the stability set of the third kind of ~MFP(a, b), which denoted by Ss(d, b, x*, 6), is defined by
S3(a,b,x*,6) = {(a,b)~nl Ili~(x*,a,b) - ~(~, d, b)ll < 6}, where ~(x, a, b) = (Fl(x, al, bl), F2(x, a2, b2) . . . . . Fk(X, ak, bk)) T. Remark 8. The stability set of the third kind of aMFP(a, b) is the set of all parameters corresponding to an a-pareto optimal solution ~(~i, b) of the problem aMFP(a, b), for a level-a, and any other feasible point
x*(a, b) ~ M. Theorem 7. The set Ss(~i, [~, x*, 6) is convex.
Proof. Let (a 1, bl), (a 2, b 2) e Sa(d, b, x*, 6), then
I I ~ ( x , a , b ) - ~(x,a,b)l[ <6,
for2E[0,1],
we get (1 - 2)Ili~(x*, a x, b 1) - ~(~, d, b)lJ < (1 - 2)6, 2 IIi~(x*, 0 2, b 2) - ~(~, a, b)II < ~6,
2 ~ l-0, 1],
299
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301 hence II~(x*, (1 - ~.)a x + ~.a2, (1 -- 2)b x + 2b 2) - ~(X, ti, b)II ~<(1 - 2 ) l l ~ ( x * , a 1,b 1 ) - ~(x, ti, b)l[ +~ll~(x*,a 2,b 2 ) - ~(x, ti, b)[I <6, then ((1 - 2)a 1 + 2b 2, (1 - 2)a 1 + 2b 2) s S3(ti, b, x*, 6), and hence the result follows.
[]
The determination of subset from the set Ss(ti, b, x*, 6) is given as follows: If S3(d, b, 6) # 0, then we have (a, b) ~ S3(ti, b, 6) such that I l q S ( x * , a , b ) - qS(~,d,b)ll < 6
for6>0,
and c* =fi(x*)/g~(x*), ci =f~(x)/g/(x), i = 1, 2 . . . . , k such that II~(x*, a, b) - ~(~, ti, b)II = IIY(x*, a, b, c*) - Y(~, d, fg, F~)I[ = IlaZ(x*, c*) - ~Z(~, g)II = I]&(Z(x*, c*) - Z(~, ?.)) + (a - ~)Z(x*, c*)I1 ~< II&II liE(x*, c*) - L(~, e)II + Ila - a II liE(x*, c*)II < hence IIcr - &ll <
6 -
II&ll IlL(x*, c * ) - L ( ~ , c)ll
IlL(x*, c*)II
= e,
where Y (x, a, b, c) = (Zl (x, al, bl, cl), Z2(x, a2, b2, c2), ... ,Zk(X, ak, bk, Ck))T, L(x, c) = (fl (x) - cl gl (x), fE(x) - c2g2(x) . . . . . fk(x) - Ckgk(X)) x, fi(x, ai) = aifi(x), gi(x, bi) = bigi(x)
and
oi = (al/bi),
i = 1, 2, . . . , k .
If N l ( x * ) denoted the sets N I ( x * ) = {(a, b) ~ R2klll(a/b ) -- (d/b)II < e}, then N l ( x * ) c S3(ti, b, 6). In order that N~(x*) ~ O, then it is clear that either ~ is large or IlL(x, c) - L(Y~, g)II is sufficiently small. Remark 9. It must be noted that N l ( x * ) c $1(£).
7. The stability set of the fourth kind Definition 9. Given a certain (ti t, b t) E B with a corresponding ~-pareto optimal solution )~ of the problem ~MFP(a, b) and 6 > 0, then the stability set of fourth kind of ~MFP(a, b), which denoted by S4(~i, b, 6), is defined by S4(ti, b, 6) = { ( a , b ) e B I 3 x e M, such that II~(x,a,b) - ~(2, ~, b)ll < 6}.
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Remark 10. The stability set of the fourth kind of aMFP(a, b) is the set of all parameters corresponding to an a-pareto optimal solution ~(& b) and 6 > 0 such that there exist x(a, b) • M and the difference between objective functions value is smaller than or equal to 6. L e m m a 4. The set S4(& b, 6) is convex. The proof follows as the proof of Theorem 8. Theorem 8. The set S4(d, b, 6) is closed in x. Proof. Let x*(a*, b*) • S4(d, b, 6), for (a*, b*) • S~(~, b), n = 1, 2, ... be sequence of points which converges to x*(a*, b*), then,
II~(x*,a*,b*)- ~(~, d, b)ll < 6 and
II~(x*~,a*,b*) - ~(x*,a*,b*)ll --, 0 as n ~ ~ . Therefore, II~(x*, a*, b*) - ~(~, d, b)II = II~(x*, a*, b*) - ~(x*, a*, b*)II + II~(x~*, a*, b*) - ~(g, d, b)II ~<~(x*,a*,b*)
-
~ ( x * , a * , b * ) l l + l l ~(xn,a * *, b * ) -
which means that x*(a*, b*) • S4(d, [~, 6), and hence the result follows.
~(~,d,b)ll < 6,
[]
Remark 11. Ss(d, b, x*, 6) c S4(d, b, 6). Remark 12. If N2 = Ux~u Nl(x*), then N2 c S4(d, b, 6). We can deduce N2 as in the following case. If L(x, c) is a contraction mapping on M, i.e., there exist a proper fraction ~ such that IlL(x*, c*) - Z(£, e)II ~< I{x* - £ I1(~1 + ~2)IIc* II + IIg(~)II I1~ - c* II ~< ~ IIx* - ~ II, where ~1 and ~2 are the proper fraction corresponding to the contraction mappingf~ and gi, respectively, then using Cauchy's inequality, it follows that Ila - ~11 <
6 - ~ll~ll Ilx* - 411 = y(x). HE(x*, c*)II
If .N1 denoted to the set
Nx (x*) = {(at, bt) • ESkl II(a/b) - (d/b)II < y(x)}, then N2 = Ux~u Nx(x*).
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
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8. Conclusion
In this paper, we investigated the stability of multiobjeetive nonlinear fractional programming problem with fuzzy parameters in the objectives functions. These fuzzy parameters are characterized by fuzzy parameters, through the use of the concept of ~-pareto optimal solutions. For the nonfuzzy problem a qualitative analysis of the solvability set of parameters, the stability set of the first, the second, the third and the fourth kind are given. The determination of the stability set of the third and fourth kind are presented.
References [1] E.E. Ammar, M.A. Kassem, Stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints, Fuzzy Sets and Systems 74 (1995) 343-351. [2] V. Chankong, Y.Y. Haimes, Multiobjective Decision Making Theory and Methodology, North-HoUand Series in System Science and Engineering, New York, 1985. [3] W. Dinkelback, On nonlinear fractional programming, Mang. Sci. 43 (1967) 492-498. [4] A. E1-Banna, A study on parameteric multiobjective problems without differentiability, Comp. Math. Appl. 26 (12) (1993) 87-92. [5] M. Grundspan, Fractional programming: a survey, Technical Report 50, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 1971. [.6] J. Guddat, F. Guerra, K. Tammer, K. Wendler, Multiobjective and Stochastic Optimization Based on Parametric Optimization, Akademie-Verlage, Berlin, 1985. [7] T. Ibaraki, Parametric approaches to fractional problems, Math. Prog. 26 (3) (1983). [.8] E. Ibrahim, M. Osman, M. Ragab, An algorithm for solving multiobjective nonlinear fractional problems, 8th European Conf. on Operational Research, Lisbon, Portugal, 16-19 September (1989). [.9] J.R. Isbell, W. Marlow, Attrition games, Naval Res. Logist. Quart. 3 (1956) 71-93. [10] A. Kaufman, M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, Elsevier Science Publishers, Amsterdam, 1988. [11] P.Q. Khanh, Proper solution of vector optimization problems, JOTP 74 (1992) 1. [,12] S. Orlovski, Multiobjective programming problems with fuzzy parameters, Control Cybernet. 13 (3) (1984) 175-183. [,13] M. Osman, Qualitative analysis of basic notions in parametric programming, II (parameters in the objective function), Apl. Mat. 22 (1977) 333-348. [14] M. Osman, A. EL-Banna, Stability of multiobjective nonlinear programming problems with fuzzy parameters, Math. Comput. Simulation 35 (1993) 321-326. [,15] P.M. Pardalos, An algorithm for a class of nonlinear parametric fractional problems using ranking of the vertices, BIT 26 (1986) 392-395. 1-16] P.M. Pardalos, A.T. Philips, Global optimization of fractional programs, J. Global Optim. 1 (1991) 173-182. [17] M. Sakawa, H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy Sets and Systems 29 (1989) 315-326. [,18] S. Schaible, Fractional programming I, Duality, Management Sci. 22 (1976a) 858-867. [,19] H. Tanaka, K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems 13 (1) (1984) 1-10.
Fuzzy Sets and Systems99 (1998) 291-301
On some basic notions of fuzzy parametric nonsmooth multiobjective nonlinear fractional programming problems Elsaid Ebrahim Ammar Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
Received March 1996;revisedJanuary 1997
Abstract
Different basic notions like the solvability set, the stability set of the first and of the second kind for the smooth fuzzy parametric multiobjective NLP programs have been discussed in several papers. In this paper, these notions have been defined and qualitatively analyzed for a general class of nonsmooth multiobjective nonlinear fractional programming problems with fuzzy parameters under the concept of ct-pareto optimality. The stability sets of the third and of the fourth kinds have also been defined and analyzed for this problem. A determination of the stability sets of the third and of the fourth kinds will be given. © 1998 Elsevier Science B.V. All rights reserved Keywords: Multiobjective nonlinear fractional programming; Nonsmooth functions; Parametric approach; Fuzzy para-
meters; ~t-pareto optimality; Stability
1. Introduction
Application and algorithms for fractional programming have been treated in considerable detail since the early work of Isbell and Marlow 1,9]. Among the many applications are the portfolio selection, stock cutting, game theory and numerous decision problems in management science. See Grundspan r5], Schaible 1,18], and Pardalos and Philips [16] for the most recent surveys. In earlier work, Osman 1,13] introduced the notions of the solvability set, stability set of the first kind and stability set of the second kind, and analyzed these concepts for parametric convex nonlinear programming problems. Recently, Tanaka and Asai [19] formulated multiobjective linear programming problems with fuzzy parameters. Orlovski [12] formulated general multiobjective parameters. Sakawa and Yano [16] introduced the concept of ~-pareto optimality of fuzzy parametric programs. In [1] we studied a multiobjective nonlinear programming problem with fuzzy parameter in the constraints functions. Also Osman and E 1Banna [14] introduced an algorithm for obtaining the subset of the parametric space which has the same corresponding ct-pareto optimal solution. In this paper, a qualitative analysis of the notions the solvability set, the stability sets of the first kind and of the second kind of the multiobjective nonlinear fractional programming problems (FMFP) with fuzzy parameters in the ratio of the objective functions without differentiability assumptions are presented. 0165-0114/98/$19.00 © 1998 ElsevierScienceB.V. All rights reserved PII S0165-0114(97)00033-X
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
292
A parametric analysis of the equivalent nonfuzzy (eMFP) problem (for a level-s) of this problem is given. A determination of the stability sets of the third and of the fourth kinds of e M F P problem will be introduced.
2. Problem formulation
Consider the following fuzzy parametric multiobjective nonlinear fractional programming problems: FMFP(~i,b)
min
Fl(x, 41,bl) - f x ( x ' 4 1 ) gl(x, bx) F2(x ' 42 , t~2) __ f2(X, 42) g2(x, bE)
Fk(x, ak, bk) - A(x, a,) gk(x, b,)' s.t.
x~M={x~R"lhj(x)<~O,j=l,
2 . . . . . m},
wherefi(x, ai) and gi(.~, bi), i = 1, 2, ..., k are nonsmooth convex and concave functions on E"+k,f/(x, 4i) >~ 0 and gi(x, bi) > 0 for each i = 1, 2. . . . . k, respectively. ~ii = (41,42, ... ,ak), /~i = (bl, b2. . . . . bk) represents the vectors of fuzzy parameters in the objectivesj~(x, 4~, bk), i = 1, 2. . . . . k. These fuzzy parameters are assumed to be characterized as the fuzzy numbers introduced in [4]. The functions hi(x), j = 1. . . . , m are nonsmooth convex functions on ~". Definition 1 (Sakawa and Yano [17]). The s-level set of the fuzzy numbers 41, bl are defined as the ordinary set S~(ti, b) for which the degree of their membership functions exceeds the level e: S~(&/~) = {(a,b) e 1~2kll~,(ai) >>.e, #b,(bi) >~ e, i = 1,2 . . . . . k}. For a certain degree of e, the problem (FMFP) can be written in the following nonfuzzy form [16]: eMFP(a, b) min s.t.
(Fl(x, al, bl), F2(x, a2, b2) . . . . . Fk(X, ak, bk)) T X e M and (a, b) e S,(4, b).
In this paper, we assume that the problem eMFP(a, b) is stable [2, 14]. Definition 2. A point ~(a, b) ~ M is said to be an e-pareto optimal solution of the eMFP(a, b) problem if and only if for at least one (d, b) s S~(&/~), there does not exist another x(a, b) ~ M such that
F,(x, d,, b,) - fi(x, d~) <~fi(2, d~) _ F,(2, d,, bi), gi(x, bi) gi(x, bi)
i = 1, 2, ..., k
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
293
with strict inequality for at least one i e {1, 2, ..., k } holds. In [3] Dinkelback shows that the solution of problem aMFP(a, b) is a solution of the following auxiliary problem ctMFP(a, b, c): ~tMFP(a, b, c)
min
Z~(x, a~, b~, c~) =f~(x, a~) - c~g~(x, b~),
min
Z~(x, a2, b2, c2) =f2(x,
min
Zk(X, ak, bk, Ck) = fk(X, ak) -- Ckgk(X, bk),
s.t.
X e M, (a, b) ~ S~(t~,b),
a2) --
C2g2(X, b2),
where c~ ~ ~k, i = 1, 2, ..., k are nonnegative real numbers. The range [_ci,?i] of ci, i = 1, 2 . . . . , k can be determined from the set of pay-off value of the problem ~tMFP(a, b, c) as in I-8]. Definition 3. A point ~(a, b, c) ~ M is said to be an ~-pareto optimal solution of ctMFP(a, b, c) if and only if for at least one (~i~, b~) ~ S,(ti, b), ~ ~ ~k and ~ >/0, i = 1, 2, ..., k there does not exist another x(a, b, c) ~ M such that
Z~(x, ~, b~, ~) ~< Z~(~, 81, b~, ~)
for i = 1, 2, ..., k
with strict inequality for at least one i ~ { 1, 2 . . . . . k) holds. The above problem ~MFP(a, b, c) can also be reformulated in the weighting problem P(a, b, c, co) as in the following form: k
P(a, b, c, co) rain
~ coiZi(x, ai, bi, ci) i=1
s.t.
x ~ M,
(a, b) ~ S,(8, b),
where O = {~i/> 0, i = 1, 2. . . . , k, coi ~ 0, E~= ~ co~ = 1}. Definition 4. A point ~(a, b, c, co) ~ M is said to be an optimal solution of the P(a, b, c, co) problem if and only if for at least one (ti, b) ~ S,(8, b), 3i ~ ~k, e3i >/ 0, i = 1, 2. . . . ,k, and o3i :~ 0, we have k
k
i=1
i=1
It is well known from [2, 7] that the ct-optimal solution x(2, #, v, co) of the weighting problem P(a, b, c, co) is an 0t-pareto optimal solution of~MFP(a, b, c) problem if and only if at least one of the following conditions are valid: (1) Problem P(a, b, c, co) has a unique solution. (2) All the weights of the weighting problem concerned are strictly positive. Let us consider the following sets: E(a, b) = {~ ~ •" 1~ is ~-pareto optimal solution of 0tMFP(a, b)}, E(a, b, c) = {~ e Rnl~ is ~t-pareto optimal solution of ~MFP(a, b, c)}, and E(a, b, c, co) = {x e ~nlx is an optimal solution of P(a, b, c, co)}, are the set of ~-pareto optimal solutions of the ~MFP(a, b), ~MFP(a, b, c) and the set of or-optimal solutions of P(a, b, c, co), respectively.
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
294
3. The solvability set
Definition 5. The solvability set B of problem aMFP(a, b) is defined by B = {(a, b) • R2kl Problem (aMFP) has the a-pareto optimal solution}. Remark 1. B = {(a, b) • R2k[ E(a, b) ~ O}. Lemma 1. If E(a, b, c, o9) is a bounded set, then B is a convex set. Proof. Suppose that (~V, b ~) • B, e = 1, 2 with a corresponding to a-level of the problem aMFP(a, b), then there exist 5 , , i = 1 , 2 . . . . . k are nonnegative real numbers with Z~(~,de, be, d,)=O and o3,~>0, i = 1, 2 . . . . . k, 03~ ~ 0 such that
(-SiZi(x, al, b~, el) <~ ~ (-SiZi(x, a~, b:, ci), i=1
i=1
and
~, cfiZ,(~, a 2, b 2, 5,) <~ ~, (SiZi(x, a 2, b 2, ci), i=1
i=1
for all x(a, b) • M, (a, b) • S~(5, b). Therefore, for 0 ~< 2 ~< 1 and by the convexity of Z,, i = 1, 2 . . . . , k, we have
~iZ,(~, dti, bi, ~) <~ i=1
(SZi(x, di, bi, 5,), ,=1
for all x(tl, 6) • M, where ai = (1 - 2) a/1 + 2a 2 and/~ = (l - 2)b I + 2b 2, then (~,,/~,) • B, and therefore B is a convex set. []
4. The stability set of the first kind
Definition 6. Given a certain (d, b) with a corresponding a-pareto optimal solution ~ of the problem aMFP(a, b), then the stability set of the first kind $1 (~, d, b) of aMFP(a, b) corresponding to ~ is defined by Sl(g, 5, b) = {(a, b ) • R2kl~ is an a-pareto optimal solution of problem aMFP(a, b)}. Remark 2. The stability set of the first kind of the aMFP(a, b) problem is the set of all parameters corresponding to one a-pareto optimal solution. Theorem 1. The set $1 (~, ti, b) is convex.
Proof. If $1(~, d, b) is empty or a single-point set, then the proof is trivial. Suppose that (a 1, bl), (a 2, b 2) • $1(~, d, b), then there exist g, >~ 0 and aS, t> O, i = 1, 2, ..., k, o3, ¢ 0 such that k
Y. i=1 k
k
al ) -
bl ) ] <. Y.
)-
/=1 k
a3,[j](~, a 2) - 5,g,(~, b~)] ~< ~ 03i[A(x, a 2) - 6,g,(x, b2)], i=1
i=1
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
295
for x e M, a n d Zi(~, a~, be, 8~) = 0, e = 1, 2, i = 1, 2, ..., k. F r o m the convexity ofJ~, g~ in a, b if 2 e [0, 1] a n d ~i = 2a/~ + (1 - 2)a 2, b, = 2b~ + (1 - 2)b~, we get k
k
a , Ef,(~, a,)
-
e,g,(~, bi)] ~< ~] o3,[A(x, at) - ¢,o,(x, ~,,)]
i=1
i=1
a n d Zi(~, ~i, bi, ~) = 0, i = 1, 2, ..., k, i.e. (~, b) e S~ (~, 8, b), a n d hence the result follows.
[]
Theorem 2. The set S~(~, ~, [~) is closed. Proof. Assume that the sequence (a t"), b ¢n~)~ S1(~, ti, b), tai " ~) , oi ,tn),~ ~ ~ai , (o), b~°)) as n -+ o~, then there exist ~, ~ ~*, ~ t> 0 a n d O) i ~ O, i = 1, 2, ..., k, e31 4: O, such that k
k
e3i[f/(~, al n)) -- ~,gi(~, bln))] ~< ~ o3[f~(x, al n)) - ~ig,(x, bln))] i=1
i=1
a n d Zi(x,- a~tn) , ,in) - , = 0. By taking the limit as n --, oo, we get o~ , co k
o3,[f~(.~, ai(o)~, - -
k
8ig,(#,
b]°))] ~< ~
i=l
and
~,[f~(x,
al °)) - c',#i(x, bl°))]
i=1
Z / ' . ~ ~(0) L(O) - x
i~ , -i , vi , co = O, then (a t°), b t°)) e S~(~, ti, b) a n d hence the result follows.
[]
T h e o r e m 3. I f Ft(x, ai, bi), i = 1, 2, ..., k are strictly convex functions on M, then
Sl(x, d, b)nSl(x*, a*, b*) = O, where f~, x* are two distinct o~-pareto optimal solutions of 0tMFP(a, b). Proof. Let Sl(~, a, b)c~Sl(x*, a*, b*) ~ O, then (a, b) e S l ( X , a , b ) ( " 3 S l ( x * , a*, b*), there exist m o r e t h a n one ~-pareto optimal solution for ~ M F P ( a , b), then we have the following two systems:
and
[J;(x, ai) - e~gi(x, b~)] - [A(x*, a~) - ~o~(x*, b,)] ~< 0 with strict inequality for at least one i, have no solution x(a, b)~ M, (a, b ) e S,(a,/~). Since Fi(x, a~, bO, i = 1, 2, ... ,k are strictly c o n v e x functions, then there eixst ~ >I 0 and r/~ >t 0, i -- 1, 2 . . . . ,k such that
~,[Y;(x, a,) - e,g,(x, b,)] >/~,[f,(~, a,) - ~,g,(~, b,)] and
ni[f~(x, ai) - e~gi(x, b~)] >/~i[f~(x*, ai) - e~o~(x*, b~)]. F o r 2 ~ [0, lJ, we have (1 - 2 ) ¢ ~ [ y ; ( x ,
ai) - ~ o i ( x , b~)l/> (1 - ,~00¢m,[J](~, a~) - e~o~(~, b~)]
>/(1
--
,,1.)~ir/i[J;(.,q, ai) -- cigi(.x, bi)]
- 2cigi(x, hi),
296
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
and
2~,r/,[f~(x, a~) - e,0,(x, b~)/> 2~,ni[jS(x*, a~) - eig~(x*, b~) >/,t~,tt,[jS(x*, 43 - e~o,(x*, b~)] - (1 -
,~)¢ig~(x*, [,,).
Consequently, (irh[f~(x, d~) - 2?~g~(x, b,)] ~> (it/~[f~().2 + (1 - 2)x*, di) -- 2~g~(2# + (1 -- 2)x*, bi)].
Putting k
k
then 03, > O, ~ 03, = 1
(,rh/~,, i = 1, 2, ..., k,
(,r/i = i=l
i=l
and k
k
03,Ef~(2~ + (1 - 2)x*, ~i~)- 2~g,(2ff + (1 - 2)x*, b,)],
03/Eli(x, gt,) - 2e, gi(x, b,)]/> i=1
i=l
which leads to a contradiction, hence the result follows.
[]
5. The stability set of the second kind Definition 7. Given a certain (ti, b) c B with a corresponding ~-pareto optimal solution ~ of the problem 0~MFP(a, b), I c {1, 2 . . . . ,k}, J c {1, 2 . . . . ,r} and ~ c 0(& b; (I, J)), where 69(4, b; (I, J)) = {(x, a, b) c R"+Zkll~n,(al) = ~, #b,(bi) = ~, i ~ I _ {1, 2 . . . . , k}, hi(x) = O,j ~ J c_ {1, 2. . . . . r} and pn,(a~) > ~, pb,(bi) > ~, i•I, hi(x) < O, jq~J},
then the stability set of the second kind S2(0(a, b; (I, J))) of the problem ~MFP(a, b) is defined by Sz(0(& b; (I, J))) = {(a, b) 6 B I 0(ti, b; (I, J)) contains an a-pareto optimal solutions of problem (aMFP)}. Remark 3. The stability set of the second kind of the problem ~MFP(a, b) represents the set of all parameters (a, b) c R 2k for which corresponding to each one of them there exist an a-pareto optimal point such that all this a-pareto optimal points either rest on one side of the restriction set M or it is interior, in other words, it is the set of all parameters corresponding to one side of the feasible region M if this side contains an a-pareto optimal solution of ~MFP(a, b). Remark 4. It follows from Definitions 6 and 7 that S2(0(d, b; (I, J))) = U $1(~', d, b), ,Eli
where I' = {i1 ~i ~ O(& b; (I, J)) is an ~-pareto optimal solution of aMFP(a, b) problem}.
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
297
Remark 5. From Definition 6, it follows that
s2(o-(a, b; (I, J))) = [J s2(e(a, b; (I, J))), JEA
where A = { j [ J <<.Jj and 0-(5, b; (I, J)) is the closure of 0(a, b; (I, J)}. Theorem 4. l f Fi(x, ai, bi), i = 1, 2 . . . . . k are strictly convex function, 11 ~ Iz, and J1 ~ J2, then
$2(Q(4 , b; (11, J1))t"~S2(Q(4, b; (•2, J2)) : 0. Proof. Suppose that (a, b) c S2((~)(d, b; (11, J1))c~Sz(p(4, b; (12, J2)), then E(a, b, c, co)rip(4, b; (11, J1)) :~ 0 and E(a, b, c, co)n0(d, b; (12, J2)) # 0. Since E(a, b, c, co) is a single point and 11 # I2, Jx # J2 then the above two relation tend to a contradiction, hence the result follows. [] Lemma 2. Under the assumption of the above theorem, S2(0(ti, b; (I, J))) is disconnected. The proof follows from Theorem 4 and Remark 4. Lemma 3. I f I' is a finite set, then $2(0(4, b; (1, J)))u{0, 0} is a closed convex set. The proof follows from Theorem 2 and Remark 4. Remark 6. A set T is said to be star shaped if there exists a point £ c T such that for all x c T, the closed line segment [2, x] e T, and £ is said to be a point of common visibility. Theorem 5. The set $2(0(& b; (I, J)))w{0, 0} is star shaped, with the point (a, b) = (0, O) its common visibility point. Proof. Let (a, b) e $2(0(4, b; (I, J))), then from Remark 3, (a, b) e $1(£ i, 4, b) for at least one index i e 1'. Then for each ~ > 0, we get (~a, eb) = e(a, b) c S2(p(d, b; (I, J))), and hence the result follows. []
Remark 7. From Definition 7 it follows that
82(Q-(4 , b; (1, g))) : (J S2(Q(~ i, bi; (1, J))), i~A
where A = {(i,j)lI <~ Ii, J <~ Ji and Q-(fi, b; (I, J)) is the closure of Q(& b; (I, J)}. Theorem 6. Under the f r s t assumption of Theorem 5 and if $2 (0 (ti, b; (I, J))) = B w {0, 0}, then
S2(p(a, [~; (I, J))) = S:(O-(4, [~; (I, J)))~J{0, 0}, where $2 (p-(5, b; (I, J))) and $2 (P(a, b; (I, J))) are the boundary and the closure of S2(Q(5, b; (1, J))), respectively. Proof. If either $2(0(5, b; (I, J))) is closed or $2(Q(4, b; (1, J))) = $1(2, 4, b), then the result is clear. Let (4, b) be a boundary point of $2 (~)(5, b; (I, J))), if (a, b) = (0, 0), also the result is clear. Otherwise, choose a sequence
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
298
(a ¢"), b¢n)) >/(0, 0), which converge to (a <°), b ~°)) as n ~ oo then there exist, 6~/> 0, 05~ ~> 0, i = 1, 2, ..., k. 05~ ¢ 0 such that (a ~n),b ~"~)~ SE(~(a, b; (I, J))) with corresponding to ~-pareto optimal solution x ~") ~ 0(~i, b; (I, J)), then k
k
~. 05, Ej'~(x~n', al n))
-
(.,g,(x ~n),bl"))] ~< ~ 05,Eft(x, al n))
i=1
-
(~9,(x, bln))]
i=1
and Z~(x ~n),u~-~n),~,h!n), 6~) = 0 for all x ~ M and for all n. By taking the limit as n ~ oo and from the finiteness of the sum and the continuity o f f and 9~, it follows that 05, f i=1
lim x ("), al °) - ~g~ lim x ("~, bl °) \n'-* ~
n-~°°
~< ~ 05, [j~(x,
- ~g,(x, bl°))]
i=1
and Z~ (limn-. ~o x t"), al °), ~,h!°),ci) = 0 for all x(a, b) ~ M, i.e., k
k
o3, Ef/(x ~°), al °)) - (.,gi(x ~°), b]°))] ~< ~ co, Efi(x, a] °)) - (~igi(x, bin))] i=I
i=1
and Z~(xt°),al °), bl °), c~)= 0 where lim, o~ x t") = x t°) exist since Bw{0,0} is closed and it is a-pareto optimality by the fact that the E(a t°~, b t°), g, 05) = {xt°)}. Therefore x ~°) E o(a ~°), bt°); (I, J)) and this completes the proof. []
6. The stability set of the third kind
Definition 8. Given a certain (d, b) ~ B with a corresponding a-pareto optimal solution ~ of the problem aMFP(a, b), x*(a, b) ~ M is any feasible point for a level-a, and 6 > 0; then the stability set of the third kind of ~MFP(a, b), which denoted by Ss(d, b, x*, 6), is defined by
S3(a,b,x*,6) = {(a,b)~nl Ili~(x*,a,b) - ~(~, d, b)ll < 6}, where ~(x, a, b) = (Fl(x, al, bl), F2(x, a2, b2) . . . . . Fk(X, ak, bk)) T. Remark 8. The stability set of the third kind of aMFP(a, b) is the set of all parameters corresponding to an a-pareto optimal solution ~(~i, b) of the problem aMFP(a, b), for a level-a, and any other feasible point
x*(a, b) ~ M. Theorem 7. The set Ss(~i, [~, x*, 6) is convex.
Proof. Let (a 1, bl), (a 2, b 2) e Sa(d, b, x*, 6), then
I I ~ ( x , a , b ) - ~(x,a,b)l[ <6,
for2E[0,1],
we get (1 - 2)Ili~(x*, a x, b 1) - ~(~, d, b)lJ < (1 - 2)6, 2 IIi~(x*, 0 2, b 2) - ~(~, a, b)II < ~6,
2 ~ l-0, 1],
299
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301 hence II~(x*, (1 - ~.)a x + ~.a2, (1 -- 2)b x + 2b 2) - ~(X, ti, b)II ~<(1 - 2 ) l l ~ ( x * , a 1,b 1 ) - ~(x, ti, b)l[ +~ll~(x*,a 2,b 2 ) - ~(x, ti, b)[I <6, then ((1 - 2)a 1 + 2b 2, (1 - 2)a 1 + 2b 2) s S3(ti, b, x*, 6), and hence the result follows.
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The determination of subset from the set Ss(ti, b, x*, 6) is given as follows: If S3(d, b, 6) # 0, then we have (a, b) ~ S3(ti, b, 6) such that I l q S ( x * , a , b ) - qS(~,d,b)ll < 6
for6>0,
and c* =fi(x*)/g~(x*), ci =f~(x)/g/(x), i = 1, 2 . . . . , k such that II~(x*, a, b) - ~(~, ti, b)II = IIY(x*, a, b, c*) - Y(~, d, fg, F~)I[ = IlaZ(x*, c*) - ~Z(~, g)II = I]&(Z(x*, c*) - Z(~, ?.)) + (a - ~)Z(x*, c*)I1 ~< II&II liE(x*, c*) - L(~, e)II + Ila - a II liE(x*, c*)II < hence IIcr - &ll <
6 -
II&ll IlL(x*, c * ) - L ( ~ , c)ll
IlL(x*, c*)II
= e,
where Y (x, a, b, c) = (Zl (x, al, bl, cl), Z2(x, a2, b2, c2), ... ,Zk(X, ak, bk, Ck))T, L(x, c) = (fl (x) - cl gl (x), fE(x) - c2g2(x) . . . . . fk(x) - Ckgk(X)) x, fi(x, ai) = aifi(x), gi(x, bi) = bigi(x)
and
oi = (al/bi),
i = 1, 2, . . . , k .
If N l ( x * ) denoted the sets N I ( x * ) = {(a, b) ~ R2klll(a/b ) -- (d/b)II < e}, then N l ( x * ) c S3(ti, b, 6). In order that N~(x*) ~ O, then it is clear that either ~ is large or IlL(x, c) - L(Y~, g)II is sufficiently small. Remark 9. It must be noted that N l ( x * ) c $1(£).
7. The stability set of the fourth kind Definition 9. Given a certain (ti t, b t) E B with a corresponding ~-pareto optimal solution )~ of the problem ~MFP(a, b) and 6 > 0, then the stability set of fourth kind of ~MFP(a, b), which denoted by S4(~i, b, 6), is defined by S4(ti, b, 6) = { ( a , b ) e B I 3 x e M, such that II~(x,a,b) - ~(2, ~, b)ll < 6}.
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E.E. .4mmar / Fuzzy Sets and Systems 99 (1998) 291-301
Remark 10. The stability set of the fourth kind of aMFP(a, b) is the set of all parameters corresponding to an a-pareto optimal solution ~(& b) and 6 > 0 such that there exist x(a, b) • M and the difference between objective functions value is smaller than or equal to 6. L e m m a 4. The set S4(& b, 6) is convex. The proof follows as the proof of Theorem 8. Theorem 8. The set S4(d, b, 6) is closed in x. Proof. Let x*(a*, b*) • S4(d, b, 6), for (a*, b*) • S~(~, b), n = 1, 2, ... be sequence of points which converges to x*(a*, b*), then,
II~(x*,a*,b*)- ~(~, d, b)ll < 6 and
II~(x*~,a*,b*) - ~(x*,a*,b*)ll --, 0 as n ~ ~ . Therefore, II~(x*, a*, b*) - ~(~, d, b)II = II~(x*, a*, b*) - ~(x*, a*, b*)II + II~(x~*, a*, b*) - ~(g, d, b)II ~<~(x*,a*,b*)
-
~ ( x * , a * , b * ) l l + l l ~(xn,a * *, b * ) -
which means that x*(a*, b*) • S4(d, [~, 6), and hence the result follows.
~(~,d,b)ll < 6,
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Remark 11. Ss(d, b, x*, 6) c S4(d, b, 6). Remark 12. If N2 = Ux~u Nl(x*), then N2 c S4(d, b, 6). We can deduce N2 as in the following case. If L(x, c) is a contraction mapping on M, i.e., there exist a proper fraction ~ such that IlL(x*, c*) - Z(£, e)II ~< I{x* - £ I1(~1 + ~2)IIc* II + IIg(~)II I1~ - c* II ~< ~ IIx* - ~ II, where ~1 and ~2 are the proper fraction corresponding to the contraction mappingf~ and gi, respectively, then using Cauchy's inequality, it follows that Ila - ~11 <
6 - ~ll~ll Ilx* - 411 = y(x). HE(x*, c*)II
If .N1 denoted to the set
Nx (x*) = {(at, bt) • ESkl II(a/b) - (d/b)II < y(x)}, then N2 = Ux~u Nx(x*).
E.E. Ammar / Fuzzy Sets and Systems 99 (1998) 291-301
301
8. Conclusion
In this paper, we investigated the stability of multiobjeetive nonlinear fractional programming problem with fuzzy parameters in the objectives functions. These fuzzy parameters are characterized by fuzzy parameters, through the use of the concept of ~-pareto optimal solutions. For the nonfuzzy problem a qualitative analysis of the solvability set of parameters, the stability set of the first, the second, the third and the fourth kind are given. The determination of the stability set of the third and fourth kind are presented.
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