(F , b, φ, ρ, θ)-UNIVEX n-SET FUNCTIONS

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IN MULTIOBJECTIVE FRACTIONAL SUBSET PROGRAMMING INVOLVING GENERALIZED

(F , b, φ, ρ, θ)-UNIVEX n-SET FUNCTIONS

ALINA PARASCHIV, LILIANA DOGARU and EUGENIA PANAITESCU

We consider some types of generalized convexity and discuss duality results for a fractional programming problem involvingn-set functions.

AMS 2000 Subject Classification: 90C29, 90C30.

Key words: multiobjective programming,n-set function, duality, generalized convexity.

1. INTRODUCTION

In this paper we present a number of semiparametric duality results under various generalized (F, b, φ, ρ, θ)-univexity hypotheses for the multiobjective fractional subset programming problem

(P) min

F₁(S)

G1(S),F₂(S)

G2(S), . . . ,F_p(S) Gp(S)

subject to H_j(S)50, j ∈q, S∈Aⁿ,

where Aⁿ is the n-fold product of the σ-algebra A of subsets of a given set X, F is not necessarily a sublinear function, Fi, Gi, i ∈ p = {1,2, . . . , p}, and Hj, j ∈ q = {1,2, . . . , q}, are real-valued functions defined on Aⁿ , and G_i(S)>0 for alli∈p and S∈Aⁿ such that H_j(S)50, j∈q. Let F be the set of all feasible solutions of (P).

Following the introduction of the notion of convexity for set functions by Morris [7] and its extension for n-set functions by Corley [3], various gen- eralizations of convexity for set and n-set functions were proposed in Lee [4], Lin [5], Preda [8, 10], Stancu-Minasian and Preda [12], Zalmai [13, 14, 15, 16].

More precisely, quasiconvexity and pseudoconvexity for set functions were defined in [4], and for n-set functions in [5]; generalized ρ-convexity for n-set functions was defined in [13], (F, ρ)-convexity in [8, 9], (ρ, b)-vexity in [10], (F, α, ρ, θ)-V-convexity in [14, 15] and (F, b, φ, ρ, θ)-univexity in [16]. Also, in

MATH. REPORTS10(60),4 (2008), 347–358

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[2], some types of generalized convexity and optimality and duality results for a multiobjective programming problem involving n-set functions were given.

For formulating and proving various collections of duality results, we shall use the class of generalized convex n-set functions called (F, b, φ, ρ, θ)- univex functions, which was defined in [16]. Until now, F was assumed to be a sublinear function in the third argument. In our approach, we suppose that F is a convex function in the third argument, as in Preda et al. [11] and Beldiman and Paraschiv [1].

2. PRELIMINARIES AND DEFINITIONS

Let (X, A, µ) be a finite atomless measure space with L1(X, A, µ) sepa- rable, and let dbe the pseudometric on Aⁿ defined by

d(R, S) =

" _n X

k=1

µ²(R_k∆S_k)

#1/2

,

where R = (R1, . . . , Rn) and S = (S1, . . . , Sn) ∈Aⁿ and ∆ denotes the sym- metric difference. Thus, (Aⁿ, d) is a pseudometric space. For h∈L1(X, A, µ) andT ∈A, the integralR

T hdµis denoted byhh, χ_Ti, whereχ_T ∈L∞(X, A, µ) is the indicator (characteristic) function of T.

Definition 2.1 ([7]). A functionF :A→Ris said to be differentiable at S^∗ ∈A if there exist DF(S^∗) ∈L1(X, A, µ), called the derivative ofF at S^∗, and V_F :A×A→R such that

F(S) =F(S^∗) +hDF(S^∗), χ_S−χ_S^∗i+V_F(S, S^∗) for each S ∈A whereV_F(S, S^∗) is o(d(S, S^∗)), that is, lim

d(S,S⁰)→0

V_F(S,S^∗) d(S,S^∗) = 0.

Definition 2.2 ([7]). A function G : Aⁿ → R is said to have a partial derivative at S^∗ = (S₁^∗, . . . , S_n^∗) ∈ Aⁿ with respect to its ith argument if the functionF(Si) =G(S₁^∗, . . . , S_i−1^∗ , Si, S_i+1^∗ , . . . , S_n^∗) has derivativeDF(S_i^∗), i ∈ n = {1,2, . . . , n}. We define D_iG(S^∗) = DF(S_i^∗) and write DF(S^∗) = (D1F(S^∗), . . . , DnF(S^∗)).

Definition 2.3 ([3]). FunctionG:Aⁿ →Ris said to be differentiable at S^∗ if there exist DF(S^∗) andW_G :Aⁿ×Aⁿ→R such that

G(S) =G(S^∗) +

n

X

i=1

DiG(S^∗), χSi−χ_S^∗

i

+WG(S, S^∗), where WG(S, S^∗) iso(d(S, S^∗)) for allS ∈Aⁿ.

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Definition 2.4. A function Ψ :Rⁿ → R is said to be sublinear (super- linear) if

Ψ(x+y)5 (=) Ψ(x) + Ψ(y)

for all x, y∈Rⁿ, and Ψ(ax) =aΨ(x) for all x∈Rⁿ and a∈R₊.

In the following we consider F : Aⁿ×Aⁿ×Rⁿ → R. The definitions below unify the concepts of (F, ρ)-convexity, (F, ρ)-pseudoconvexity, (F, ρ)- quasiconvexity from Preda [10] and univexity, pseudounivexity, quasiunivexity from Mishra [6].

Definition 2.5 ([16]). A function F is said to be (strictly) (F, b, φ, ρ, θ)- univex at S^∗ if there exist a function b :Aⁿ×Aⁿ → R with positive values, a function θ :Aⁿ×Aⁿ → Aⁿ×Aⁿ such that S 6=S^∗ ⇒ θ(S, S^∗)6= (0,0),a function φ:R→R,and a real number ρsuch that

φ(F(S)−F(S^∗)) (>) =F(S, S^∗;b(S, S^∗)DF(S^∗)) +ρd²(θ(S, S^∗)) for each S ∈Aⁿ.

Definition 2.6 ([16]). A function F is said to be (strictly) (F, b, φ, ρ, θ)- pseudounivex at S^∗ if there exist a function b : Aⁿ×Aⁿ → R with positive values, a function θ :Aⁿ×Aⁿ → Aⁿ×Aⁿ such that S 6= S^∗ ⇒ θ(S, S^∗) 6=

(0,0),a functionφ:R→R,and a real number ρ such that

F(S, S^∗;b(S, S^∗)DF(S^∗))=−ρd²(θ(S, S^∗))⇒φ(F(S)−F(S^∗)) (>) =0 for each S ∈Aⁿ,S 6=S^∗.

Definition2.7 ([16]). A functionF is said to be (prestrictly) (F, b, φ, ρ, θ)- quasiunivex atS^∗if there exist a functionb:Aⁿ×Aⁿ→Rwith positive values, a function θ :Aⁿ×Aⁿ → Aⁿ×Aⁿ such that S 6=S^∗ ⇒ θ(S, S^∗)6= (0,0),a function φ:R→R,and a real number ρsuch that

φ(F(S)−F(S^∗)) (<) 50⇒ F(S, S^∗;b(S, S^∗)DF(S^∗)5−ρd²(θ(S, S^∗)) for each S ∈Aⁿ.

3. ZALMAI’S DUAL MODEL

In this section, as in Zalmai [16], we consider the duality model max

F₁(T)

G1(T),F₂(T)

G2(T), . . . , F_p(T) Gp(T)

(D)

subject to

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F

S, T;

p

X

i=1

u_i[G_i(T)F_i(S)−F_i(T)G_i(S)]+

q

X

j=1

v_jDH_j(T)

=0 for allS∈Aⁿ, (3.1)

q

X

j=1

v_jH_j(T)=0, j∈q, T ∈Aⁿ, u∈U₀, v∈R^q₊, (3.2)

where F(S, T;·) :Lⁿ₁(X, A, µ)→Rand U₀ =n

u∈R^p:u=0,

p

P

i=1

u_i = 1o . We suppose thatF is a convex function in the third argument. For fixed S, u and v we define the functions

f_i(T, S, u) =G_i(S)F_i(T)−F_i(S)G_i(T), i∈p, f(T, S, u) =

p

X

i=1

ui[Gi(S)Fi(T)−Fi(S)Gi(T)],

h(T, v) =

q

X

j=1

v_jH_j(T).

For given u^∗ ∈U₀, v^∗ ∈R^q₊,let I₊(u^∗) ={i∈p:u^∗_i >0}and J₊(v^∗) ={ j∈ q :v_j^∗>0}.Now, we have a first weak duality result:

Theorem 3.1(Weak duality). Let S and (T, u, v) with u >0 be arbitrary feasible solutions of (P) and (D), respectively,and assume that any one of the following sets of hypotheses is satisfied:

(a) (i) 2f(·, T, u) is (F, b, φ, ρ, θ)-pseudounivex at T and φ(a) = 0 ⇒ a=0;

(ii) 2βHb j is(F, b,φej,ρej, θ)-quasiunivex at T,φej is increasing and φej(0) = 0 for each j∈J₊ =J₊(v);

(iii) ρ+ P

j∈J+

vj

βbρej =0;

(b) (i) 2f(·, T, u) is (F, b, φ, ρ, θ)-pseudounivex at T and φ(a) = 0 ⇒ a=0;

(ii) 2h(·, v) is (F, b,φ,e ρ, θ)-quasiunivex ate T, φe is increasing and φ(0) = 0;e

(iii) ρ+ρe=0;

(c) (i) 2f(·, T, u)is prestrictly (F, b, φ, ρ, θ)-quasiunivex at T and φ(a)= 0⇒a=0;

(ii) 2bβH_j is(F, b,φe_j,ρe_j, θ)-quasiunivex at T,φe_j is increasing and φe_j(0) = 0 for each j∈J+ =J+(v);

(iii) ρ+ P

j∈J₊ vj

βbρe_j >0;

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(d) (i) 2f(·, T, u)is prestrictly (F, b, φ, ρ, θ)-quasiunivex at T and φ(a)= 0⇒a=0;

(iii) ρ+ρe=0;

(e) (i) 2f(·, T, u)is prestrictly (F, b, φ, ρ, θ)-quasiunivex at T and φ(a)= 0⇒a=0;

(ii) 2βHb j is strictly (F, b,φej,ρej, θ)-pseudounivex at T, φej is increasing and φe_j(0) = 0 for eachj∈J₊=J₊(v);

(iii) ρ+ P

j∈J₊ vj

βbρe_j =0;

(f) (i) 2f(·, T, u)is prestrictly (F, b, φ, ρ, θ)-quasiunivex at T and φ(a)= 0⇒a=0;

(ii) 2h(·, v) is strictly (F, b,φ,e ρ, θ)-pseudounivex ate T, φe is increasing and φ(0) = 0;e

(iii) ρ+ρe=0;

Then ϕ(S) ψ(T, u, v), where ψ = (ψ1, . . . , ψp) is the objective function of (D).

Proof. (a). LetS be an arbitrary feasible solution of (P). It then follows from our hypotheses in (ii) that

2βHb j(S)50 = 2βHb j(T) ⇒2βHb j(S)−2βHb j(T)50 ⇒

⇒ φej 2βHb j(S)−2βHb j(T) 50, which implies that

F S, T;b(S, T)2βDHb j(T)

5−ρejd²(θ(S, T)).

It follows from the convexity of F(S, T;·),(3.1) and b(S, T)>0 that F

S, T;b(S, T)

q

X

j=1

2vjDHj(T)

=F

S, T;b(S, T)

q

X

j=1

2v_j βb

βDHb j(T)

≤

≤ X

j∈J+

vj

βbF S, T;b(S, T)2βDHb j(T)

5−X

j∈J+

vj

βbρejd²(θ(S, T)).

From (3.1) we have F

S, T;b(S, T)

p

X

i=1

2u_i[G_i(T)DF_i(T)−F_i(T)DG_i(T)]

+

+F

S, T;b(S, T)

q

X

j=1

2v_jDH_j(T)

=0.

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Then F

S, T;b(S, T)

p

X

i=1

2u_i[G_i(T)DF_i(T)−F_i(T)DG_i(T)]

=

=X

j∈J₊

v_j

βbρe_jd²(θ(S, T))=−ρd²(θ(S, T)),

where the second inequality follows from (iii). By (i), this inequality implies that

φ(2f(S, T, u)−2f(T, T, u))=0, which, because of the properties of the function φ, reduces to

2f(S, T, u)=2f(T, T, u).

But f(T, T, u) = 0,hence f(S, T, u)=0,which is precisely

p

X

i=1

2ui[Gi(T)Fi(S)−Fi(T)Gi(S)]=0.

Suppose that ϕ(S) ≤ ψ(T, u, v). Hence _G^Fⁱ^(S)

i(S) 5 _G^Fⁱ_i^(T_(T⁾₎, with strict inequality for at least one subscript. That means

Fi(S)Gi(T)−Fi(T)Gi(S)50,

with strict inequality for at least one subscript. Multiplying by u =0, u6= 0 and then adding yield

p

X

i=1

2ui[Gi(T)Fi(S)−Fi(T)Gi(S)]<0, which is false. Therefore, we conclude that ϕ(S)ψ(T, u, v).

(b)–(f). The proofs are similar to that of part (a).

The theorem is thus proved.

Using some partitions ofI and J, we have the duality results below.

Theorem 3.2(Weak duality). Let S and (T, u, v) be arbitrary feasible solutions of(P)and(D), respectively,and assume that any one of the following sets of hypotheses is satisfied:

(a) (i) 2αfb _i(·, T) is strictly (F, b, φ_i, ρ_i, θ)-pseudounivex at T, φ_i is increasing and φi(0) = 0for each i∈I+ =I+(u);

(ii) 2bβ H_jis (F, b,φe_j,ρe_j, θ)-quasiunivex at T,φe_j is increasing and φe_j(0) = 0 for each j∈J+ =J+(v);

(iii) ρ⁰+ P

j∈J₊ vj

βbρe_j =0,where ρ⁰ = P

i∈I₊ ui

αbρ_i;

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(b) (i) 2αfb i(·, T) is strictly (F, b, φ_i, ρi, θ)-pseudounivex at T, φi is increasing and φi(0) = 0for each i∈I+ =I+(u);

(iii) ρ⁰+ρe=0;

(c) (i) 2αfb _i(·, T) is (F, b, φ_i, ρ_i, θ)-quasiunivex at T,φ_i is increasing and φ_i(0) = 0 for eachi∈I₊=I₊(u);

(ii) 2βHb _j is strictly (F, b,φe_j,ρe_j, θ)-pseudounivex at T, φe_j is increasing and φe_j(0) = 0 for eachj∈J₊=J₊(v);

(iii) ρ⁰+ P

j∈J+

vj

βbρe_j =0;

(d) (i) 2αfb _i(·, T) is (F, b, φ_i, ρ_i, θ)-quasiunivex at T,φ_i is increasing and φ_i(0) = 0 for eachi∈I₊=I₊(u);

(ii) 2h(·, v) is strictly (F, b,φe_j,ρe_j, θ)-pseudounivex at T, φeis increasing and φ(0) = 0;e

(iii) ρ⁰+ρe=0;

(e) (i) 2α fb _i(·, T) is(F, b, φ_i, ρ_i, θ)-quasiunivex atT,φ_i is increasing and φ_i(0) = 0 for eachi∈I₊=I₊(u);

(iii) ρ⁰+ P

j∈J₊ vj

βbρej >0;

(f) (i) 2αfb i(·, T) is (F, b, φ_i, ρi, θ)-quasiunivex at T, φi is increasing and φi(0) = 0 for eachi∈I+=I+(u);

(iii) ρ⁰+ρ >e 0;

Then ϕ(S)ψ(T, u, v).

Proof. (a). Suppose that ϕ(S) ≤ ψ(T, u, v). Then there exists S ∈ F such that ϕ_i(S) 5 ϕ_i(T) for each i ∈ p and ϕ_l(S) < ϕ_l(T) for some l ∈ p.

From these inequalities and (3.2) it can be easily be seen that Gi(T)Fi(S)−Fi(T)Gi(S)50 =Gi(T)Fi(T)−Fi(T)Gi(T),

for each i∈I₊,which on account of the properties ofφ_i can be expressed as φ_i 2αb

G_i(T)F_i(S)−F_i(T)G_i(S)

−2αb[G_i(T)F_i(T)−F_i(T)G_i(T)]

50.

By (i), this implies that

F S, S^∗;b(S, T)2αb[Gi(T)DFi(T)−Fi(T)DGi(T)]

<−ρ_id²(θ(S, T))

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for each i∈I+. Since u =0, ui = 0 for each i∈p\I₊, P

i∈I+

ui = 1, using the convexity of F S, T;·

we get (3.3) F S, T; 2b(S, T)

p

X

i=1

u_i[G_i(T)DF_i(T)−F_i(T)DG_i(T)]

!

=

=F S, T; 2αb(S, Tb )

p

X

i=1

u_i

αb [G_i(T)DF_i(T)−F_i(T)DG_i(T)]

! 5

5

p

X

i=1

u_i

αbF S, T; 2αb(S, Tb ) [G_i(T)DF_i(T)−F_i(T)DG_i(T)]

<

<−X

i∈I+

u_i

αbρ_id²(θ(S, T)).

As S ∈ F, for each j ∈ J₊,we have 2βHb _j(S) 5 0 = 2bβH_j(T) and so, using the properties of φej,we get

φej

2βHb j(S)−2βHb j(T)

50, for each j ∈J₊ which by (ii) implies that

F

S, T;b(S, T)2βDHb _j(T)

5−ρe_jd²(θ(S, T)).

Because v =0, vj = 0 for each j ∈q\J₊, multiplying by vj and summariz- ing yield

q

X

j=1

v_jF

S, T;b(S, T)2bβDH_j(T) 5−

q

X

j=1

v_jρe_jd²(θ(S, T)).

Since ^v^j

βb =0,

q

P

j=1 vj

βb = 1 and F(S, T;·) is convex, we have F

S, T;

q

X

j=1

v_jb(S, T)2DH_j(T)

=F

S, T;

q

X

j=1

v_j

βbb(S, T)2βDHb _j(T)

5

q

X

j=1

vj

βbF

S, T;b(S, T)2βDHb j(T)

5−1 βb

q

X

j=1

vjρejd²(θ(S, T)), i.e.,

F

S, T;b(S, T)

q

X

j=1

vj2DHj(T)

5−1 βb

q

X

j=1

vjρejd²(θ(S, T)).

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Now, combining these relations and using the convexity of F S, T;· ,yield F

S, T; 2b(S, T)

p

X

i=1

ui[Gi(T)DFi(T)−Fi(T)DGi(T)]

+

+F

S, T; 2b(S, T)

q

X

j=1

v^∗_jDHj(T)

<−

ρ⁰+ X

j∈J+

v^∗_j βbρej

d²(θ(S, T)), which by (iii) contradicts (3.1). Hence ϕ(S)Ψ(T, u, v).

(b)–(f). The proofs are similar to that of part (a).

The proof is complete.

Theorem 3.3(Weak duality). Let S and (T, u, v) be arbitrary feasible solutions of(P)and(D), respectively, and assume that any one of the following sets of hypotheses is satisfied:

(a) (i) 3αb₁ f_i(·, T) is strictly (F, b, φ_i, ρ_i, θ)-pseudounivex at T for each i ∈ I₁₊, 3αb₂f_i(·, T) is (F, b, φ_i, ρ_i, θ)-quasiunivex at T for each i∈ I₂₊, φ_i is increasing and φ_i(0) = 0 for each i∈ I₊ = I₊(u), where {I₁₊, I₂₊} is a partition of I+,I1+ 6=∅,I2+ 6=∅;

(iii) ρ⁰+ P

j∈J₊ vj

βbρe_j =0,where ρ⁰ = P

i∈I₁₊

ui

αb1ρ_i+ P

i∈I₂₊

ui

αb2ρ_i;

(b) (i) 3αb1fi(·, T) is strictly (F, b, φ_i, ρi, θ)-pseudounivex at T for each i ∈ I₁₊, 3αb₂f_i(·, T) is (F, b, φ_i, ρ_i, θ)-quasiunivex at T for each i ∈ I₂₊, φ_i is increasing and φ_i(0) = 0 for each i ∈ I₊ = I₊(u), where {I₁₊, I₂₊} is a partition of I+,I1+ 6=∅,I2+ 6=∅;

(iii) ρ⁰+ρe=0;

(c) (i) 3αfb _i(·, T) is (F, b, φ_i, ρ_i, θ)-quasiunivex at T,φ_i is increasing and φi(0) = 0 for eachi∈I+=I+(u);

(ii) 3bβ₁H_j is strictly (F, b,φe_j,ρe_j, θ)-pseudounivex at T for each j∈J₁₊, 3βb2Hj is (F, b,φej,ρej, θ)-quasiunivex at T for each j ∈ J2+, φej is increasing and φe_j(0) = 0 for each j ∈ J₂₊, where {J₁₊, J₂₊} is a partition of J₊, J1+6=∅,J2+6=∅;

(iii) P

i∈I₊ ui

αbρ_i+ P

j∈J₁₊

vj

βb1ρe_j + P

j∈J₂₊

vj

βb2ρe_j =0;

(d) (i) 4αb1fi(·, T) is strictly (F, b, φ_i, ρi, θ)-pseudounivex at T, for each i∈I1+, 4αb2fi(·, T) is (F, b, φ_i, ρi, θ)-quasiunivex at T, for each i∈I2+, φi

is increasing and φ_i(0) = 0 for each i ∈ I₊ = I₊(u), where {I₁₊, I₂₊} is a partition of I+, I1+ 6=∅,I2+ 6=∅;

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(ii) 4βb1Hj is strictly (F, b,φej,ρej, θ)-pseudounivex at T, for each j∈J1+, 4βb₂H_j is (F, b,φe_j,ρe_j, θ)-quasiunivex at T, for each j ∈ J₊, φe_j is increasing and φej(0) = 0 for eachj∈J+,where {J₁₊, J2+} is a partition of J+;

(iii) ρ⁰+ P

j∈J₁₊

vj

βb1ρej + P

j∈J₂₊

vj

βb2ρej =0;

(iv)I1+ 6=∅ or J1+ 6=∅ or ρ⁰+ P

j∈J1+

vj

βb1ρej+ P

j∈J2+

vj

βb2ρej >0;

Then ϕ(S)ψ(T, u, v).

Proof. Suppose that ϕ(S)≤ψ(T, u, v). Proceeding as above, we obtain F

S, T; 3b(S, T)X

i∈I1+

ui[Gi(T)DFi(T)−Fi(T)DGi(T)]

+ +F

S, T; 3b(S, T)X

i∈I₂₊

u_i[G_i(T)DF_i(T)−F_i(T)DG_i(T)]

+

+F

S, T; 3b(S, T)

q

X

j=1

vjDHj(T)

<

<−

X

i∈I₁₊

ui

αb₁ρ_id²(θ(S, T)) + X

i∈I₂₊

ui

αb₂ρ_id²(θ(S, T)) + 1 βb

q

X

j=1

v_jρe_jd²(θ(S, T))

=

=−

ρ⁰+ X

j∈J+

v_j βbρej

d²(θ(S, T)), which by (iii) contradicts (3.1). Then ϕ(S)ψ(T, u, v).

Using these results, we can now prove the duality theorems below.

Theorem 3.4(Strong duality). Let S^∗ be a regular efficient solution of (P), let

F(S, S^∗;DF(S^∗)) =

n

X

k=1

D

D_kF(S^∗), χ_S_k−χ_S^∗

k

E

for any differentiable function F :Aⁿ→R and S∈Aⁿ,and assume that any one of the sets of hypotheses specified in Theorems3.1–3.3holds for all feasible solutions of (D). Then there exist u^∗ ∈U and v^∗ ∈R^q₊ such that (S^∗, u^∗, v^∗) is an efficient solution of (D) and φ(S^∗) =ψ(S^∗, u^∗, v^∗).

Proof. There existu^∗∈U andv^∗∈R^q such that (S^∗, u^∗, v^∗) is a feasible solution of (D) and φ(S^∗) = ψ(S^∗, u^∗, v^∗). It follows then from the corres- ponding parts of Theorems 3.1–3.3 that (S^∗, u^∗, v^∗) is an efficient solution of (D).

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Theorem 3.5(Strict converse duality). Let S^∗ and F(S, S^∗;·) be as in Theorem 3.4, let S,e u,e ev

be a feasible solution of (D)such that f S^∗,S,e ue 5 0, and assume that either one of the two sets of hypotheses specified in parts (a)and(b)of Theorem3.1is satisfied for the feasible solution S,e u,e ev

of(D).

Assume furthermore, that f ·,S,e ue

is strictly (F, b, φ, ρ, θ)-pseudounivex at Seand that φ(a)>0⇒a >0.Then Se=S^∗,that is,Seis an efficient solution of (P).

Theorem 3.6(Strict converse duality). Let S^∗ and F S, S^∗;·

be as in Theorem 3.4, let S,e u,e ev

be a feasible solution of (D)such that f S^∗,S,e ue 5 0, and assume that any one of the four sets of hypotheses specified in parts (c)–(f) of Theorem 3.1 is satisfied for the feasible solution S,e u,e ev

of (D).

Assume furthermore that f ·,S,e eu

is(F, b, φ, ρ, θ)-quasiunivex at Seand that φ(a)>0⇒a >0.Then Se=S^∗,that is, Seis an efficient solution of (P).

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Received 6 May 2008 “Elena Cuza” National College Str. Pestera Scarisoara 1 062071 Bucharest, Romania

“Spiru Haret” Dobrogean College Str. 14 Noiembrie 24 820009 Tulcea, Romania

and

“Carol Davila” University of Medicine and Pharmacy Faculty of Medicine

Departament of Medical Informatics and Biostatistics Bd. Eroii Sanitari No. 8

050471 Bucharest, Romania