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Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 63-73 DOI: 10.2298/YUJOR0901063S DUALITY FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS INVOLVING d -TYPE-I -SET n - FUNCTIONS I.M.STANCU-MINASIAN The Romanian Academy, Institute of Mathematical Statistics and Applied Mathematics Romania Gheorghe DOGARU “Mircea cel Bătrân”, Naval Academy Romania Andreea Mădălina STANCU The Romanian Academy, Institute of Mathematical Statistics and Applied Mathematics Romania Received: December 2007 / Accepted: May 2009 Abstract: We establish duality results under generalized convexity assumptions for a multiobjective nonlinear fractional programming problem involving d -type-I n -set functions. Our results generalize the results obtained by Preda and Stancu-Minasian [24], [25]. Keywords: d-type-I set functions, multiobjective programming, duality results. 1. INTRODUCTION Consider the multiobjective nonlinear fractional programming problem involving n -set functions 64 I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective ⎛ F (S ) Fp ( S ) ⎞ minimize F ( S ) = ⎜ 1 ,..., ⎟ ⎜ G (S ) G p ( S ) ⎟⎠ ⎝ 1 (P) subject to H j ( S ) ≤ 0, j ∈ M , S = ( S1 ,..., S n ) ∈ Γ n where Γ n is the n -fold product of a σ - algebra Γ of subsets of a given set X , M = {1, 2,..., m} , Fi , Gi , i ∈ P = {1, 2,..., p} , and H j , j ∈ M are differentiable realvalued functions defined on Γ n with Fi ( S )≥ 0 and Gi ( S ) > 0 , for all i ∈ P . (1) Let S0 = {S S ∈ Γ n , H ( S ) ≤ 0} be the set of all feasible solutions to (P), where H = ( H1 ,..., H m ) . The term “minimize” being used in Problem (P) is for finding efficient, weakly and properly efficient solutions. A feasible solution S 0 to (P) is said to be an efficient solution to (P) if there exists no other feasible solution S to (P) so that Fi ( S ) ≤ Fi ( S 0 ) , for all i ∈ P , with strict inequality for at least one i ∈ P . A feasible solution S 0 to (P) is said to be a weakly efficient solution to (P) if there exists no other feasible solution S to (P) so that Fi ( S ) < Fi ( S 0 ) , for all i ∈ P . The analysis of optimization problems with set or n -set functions i.e. selection of measurable subsets from a given space, has been the subject of several papers. For a historical survey of optimality conditions and duality for programming problems involving set and n-set functions the reader is referred to Stancu-Minasian and Preda’s review paper [28]. These problems arise in various applications including fluid flow [3], electrical insulator design [8], regional design (districting, facility location, warehouse layout, urban planning etc.) [10], statistics [11], [21] and optimal plasma confinement [30]. The general theory for optimizing set functions was first developed by Morris [20]. Many results of Morris [20] are only confined to functions of a single set. Corley [9] started to give the concepts of partial derivatives and derivatives of real-valued n -set functions. Starting from the methods used by Jeyakumar and Mond [12] and Ye [31], Suneja and Srivastava [29] defined some new classes of scalar or vector functions called d -type-I, d -pseudo-type-I, d -quasi-type-I etc. for a multiobjective nondifferentiable programming problem and obtained necessary and sufficient optimality criteria. Also, they established duality between this problem and its Wolfe-type and Mond-Weir-type duals and obtained some duality results considering the concept of a weak minimum. In particular, multiobjective fractional subset programming problems have been the focus of intense interest in the past few years, and resulted in many papers [1], [2], [4]-[7], [13]-[17], [22], [23], [28], [33]-[35]. In this paper we establish duality results under generalized convexity assumptions for a multiobjective nonlinear fractional programming problem involving I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective 65 generalized d -type-I n -set functions. Our results generalize the results obtained by Preda and Stancu-Minasian [24], [25]. 2. DEFINITIONS AND PRELIMINARIES In this section we introduce the notation and definitions which will be used throughout the paper. Let R n be the n - dimensional Euclidian space and R n+ its positive orthant, i.e. R n+ = {x = ( x j ) ∈ R n , x j ≥ 0 , j = 1,..., n} . For x = ( x1 ,..., xm ), y = ( y1 ,..., ym ) ∈ R m we put x ≤ y iff xi ≤ yi for each i ∈ M ; x ≤ y iff xi ≤ yi each i ∈ M , with x ≠ y ; x < y iff xi < yi for each i ∈ M while x y is the negation of x < y . We write x ∈ R n+ iff x ≥ 0 . Let ( X , Γ, μ ) be a finite non-atomic measure space with L1 ( X , Γ, μ ) separable, and let d be the pseudometric on Γ n defined by: 1/ 2 ⎡ n ⎤ d ( S , T ) = ⎢ ∑ μ 2 ( S k ΔTk ) ⎥ ⎣ k =1 ⎦ for S = ( S1 ,..., Sn ) , T = (T1 ,..., Tn ) ∈ Γ n , where Δ denotes the symmetric difference. Thus (Γ n , d ) is a pseudometric space, which will serve as the domain for most of the functions that will be used in this paper. For h ∈ L1 ( X , Γ, μ ) , the integral ∫ h dμ will be denoted by h, I S , where I S is S the indicator (characteristic) function of S ∈ Γ . We next introduce the notion of differentiability for n -set functions. This was originally introduced by Morris [20] for set functions and subsequently extended by Corley [9] to n -set functions. A function ϕ : Γ → R is said to be differentiable at S 0 ∈ Γ if there exist Dϕ ( S 0 ) ∈ L1 ( X , Γ, μ ) , called the derivative of ϕ at S 0 , and ψ : Γ × Γ → R such that for each S ∈ Γ , ϕ ( S ) = ϕ ( S 0 ) + Dϕ ( S 0 ), I S − I S + ψ ( S , S 0 ) , 0 where ψ ( S , S 0 ) is o(d ( S , S 0 )) , that is, ( lim 0 d S ,S ψ (S , S 0 ) )→0 d ( S , S 0 ) =0. A function F : Γ n → R is said to have a partial derivative at S 0 = ( S10 ,..., S n0 ) with respect to its k -th argument if the function ϕ ( Sk ) = F ( S10 ,..., Sk0−1 , S k , S k0+1 ,..., S n0 ) has derivative Dϕ ( S k0 ) and we define Dk F ( S 0 ) = Dϕ ( S k0 ) . If Dk F ( S 0 ) , 1 exist, then we put DF ( S 0 ) = ( D1 F ( S 0 ),..., Dn F ( S 0 )) . k n , all 66 I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective A function F : Γ n → R is said to be differentiable at S 0 if there exist DF ( S 0 ) and ψ : Γ n × Γ n → R such that n F ( S ) = F ( S 0 ) + ∑ Dk F ( S 0 ), I Sk − I S 0 + ψ ( S , S 0 ) , k k =1 where ψ ( S , S 0 ) is o(d ( S , S 0 )) , for all S ∈ Γ n . Along the lines of Jeyakumar and Mond [12] and Suneja and Srivastava [29], Preda and Stancu-Minasian [24] defined new classes of n -set functions, called d-type-I, d-quasi type-I, d-pseudo type-I, d-quasi-pseudo type-I, d-pseudo-quasi type-I. In [18] Mishra extended the generalized d -type-I vector-valued functions of Preda and Stancu-Minasian [24] to new generalized d -type-I n -set functions and establish optimality and Mond-Weir type duality results. Definition 1. [24] We say that ( F , G ) is of d -type-I at S 0 ∈ Γn if there exist functions α i , β j : Γ n × Γ n → R + \ {0} , i ∈ P, j ∈ M , such that for all S ∈ S0 , we have Fi ( S ) − Fi ( S 0 ) n ≥ α i ( S , S 0 )∑ Dk Fi ( S 0 ), I S − I S , i ∈ P 0 k k k =1 (2) and n − H j ( S 0 ) ≥ β j ( S , S 0 )∑ Dk H j ( S 0 ), I Sk − I S 0 , j ∈ M . k k =1 (3) We say that ( F , H ) is of d-semistrictly type-I at S 0 if in the above definition we have S ≠ S 0 and (2) is a strict inequality. Now, we introduce Definition 2. [32] A feasible solution S 0 to (P) is said to be a regular feasible solution if there exists Sˆ ∈ Γ n such that n H j ( S 0 ) + ∑ Dk H j ( S 0 ), I Sˆ − I S 0 < 0 , j ∈ M . k =1 k k Now, for each λ = (λ1 ,..., λ p ) ∈ R +p we consider the parametric problem minimize(F1 ( S ) − λ1G1 ( S ),..., Fp ( S ) − λ p G p ( S )) ( Pλ ) subject to H j (S ) ≤ 0, j ∈ M , S = ( S1 ,..., Sn ) ∈ Γ n . The problem ( Pλ ) is equivalent to the problem (P) in the sense that for particular choices of λi , i ∈ P , the two problems have the same set of efficient solutions. This equivalence is stated in the following lemma which is well known in fractional programming [27]. I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective 67 Lemma 3. An S 0 is an efficient solution to (P) if and only if is an efficient solution to F (S 0 ) ( Pλ0 ) with λi0 = i 0 , i = 1,..., p . Gi ( S ) In this paper the proofs of the duality results for Problem (P) will invoke the following necessary efficiency result for ( Pλ ) (see Zalmai [32], Theorem 3.2). Theorem 4. [32] Let S 0 be a regular efficient (or weakly efficient) solution to (P) and assume that Fi , Gi , i ∈ P and H j , j ∈ M , are differentiable at S 0 . Then there exist p ∑u u 0 ∈ R +p , 0 i i =1 n p k =1 i =1 = 1 , v 0 ∈ R +m , and λ 0 ∈ R +p such that ∑ ∑u 0 i m ⎛ ⎞ 0 0 0 0 0 ⎜ Dk Fi ( S ) − λi Dk Gi ( S ) + ∑ v j Dk H j ( S ), I Sk − I Sk0 ⎟ i =1 ⎝ ⎠ ≥ 0, for all , S ∈ Γn (4) u ( Fi ( S ) − λ Gi ( S ))≥ 0 , i ∈ P (5) v 0j H j ( S 0 ) = 0 , j ∈ M . (6) 0 i 0 0 i 0 3. DUALITY In this section, in the differentiable case, based on the equivalence of (P) and ( Pλ ) a dual for ( Pλ ) is defined and some duality results in d-type-I assumptions are stated. With ( Pλ ) we associate a dual stated as maximize (λ1 ,..., λ p ) (D) subject to p n ∑∑ u i =1 k =1 m 0, j =1 k =1 S ∈ Γn (7) ui ( Fi (T ) − λi Gi (T )) 0 , i∈P, 0 , j∈M , v j H j (T ) u ∈ R +p , n Dk Fi (T ) − λi Dk Gi (T ), I Sk − I Tk + ∑∑ v j Dk H j (T ) , I Sk − ITk i p ∑u i =1 i = 1 , v ∈ R m+ , λ ∈ R +p . (8) (9) (10) Let D 0 be the set of feasible solutions to (D). Let us prove the duality theorems. Theorem 5. (Weak duality) Let S and (T , u , v, λ ) be feasible solutions to problem (P) and (D), respectively such that (i1) for each i ∈ P and j ∈ M , ( Fi (⋅) − λi Gi (⋅), H j (⋅)) is 68 I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective of d-type-I at T ; (i2) ui > 0 for any i ∈ P , and for some i ∈ P and j∈M , ( Fi (⋅) − λi Gi (⋅), H j (⋅)) is of d-semistrictly type-I at T . Then for any S ∈ S0 one cannot have Fi ( S ) ≤ λi for any i ∈ P , Gi ( S ) Fj ( S ) G j (S ) (11) < λ j for some j ∈ P . (12) Proof: Let us suppose the contrary that (11) and (12) hold. Then there exists S , a feasible solution for ( Pλ ), such that (11) and (12) hold. If hypothesis (i2) holds, then ui > 0 for any i = 1,..., p . From (1), (11) and (12) we get p ∑u i =1 ( Fi ( S ) − λi Gi ( S )) < 0 . i (13) Using the feasibility of S , and the relations (9) and (10), we have v j H j ( S )≤ 0 ≤ v j H j (T ) ∀ j = 1,..., m . (14) Since α i ( S , T ) > 0, i ∈ P , and β j ( S , T ) > 0, j ∈ M , combining (8), (13) and (14) we obtain p p ui ui ∑ α (S , T ) ( F (S ) − λ G (S )) < ∑ α (S , T ) ( F (T ) − λ G (T )) i =1 i i i m v j H j (T ) i =1 β j (S , T ) +∑ i i i =1 i i i . (15) We claim that S ≠ T for if it is not true, then, from ui > 0 , i ∈ P , the feasibility of S and (8) we obtain a contradiction with (11) and (12). One the other hand, from S ≠ T , (i1) and (i2), it follows that ( Fi ( S ) − λi Gi ( S )) − ( Fi (T ) − λi Gi (T ))≥ n α i ( S , T )∑ Dk Fi (T ) − λi Dk Gi (T ), I S − IT k k =1 k (16) for any i ∈ P , with strict inequality for some i , and n − H j (T ) ≥ β j ( S , T )∑ Dk H j (T ), I Sk − ITk , j ∈ M . (17) k =1 By dividing by α i ( S , T ) > 0 and β j ( S , T ) > 0 , respectively, the above inequalities reduce to the following I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective Fi ( S ) − λi Gi ( S ) Fi (T ) − λi Gi (T ) − ≥ αi (S , T ) αi (S , T ) n ∑ k =1 69 Dk Fi (T ) − λi Dk Gi (T ), I Sk − ITk (18) for any i ∈ P , with strict inequality for some i , and H j (T ) − β j (S , T ) ≥ n ∑ k =1 Dk H j (T ), I Sk − ITk , j ∈ M (19) Multiplying the inequality (18) by ui > 0 , ∀ i ∈ P , and (19) by v j ≥ 0, ∀ j ∈ M , and summing after all i and j , respectively, yields p p ui p ui v j H j (T ) ∑ α (S , T ) ( F (S ) − λ G (S )) − ∑ α (S , T ) ( F (T ) − λ G (T )) − ∑ β i i =1 i i i p i =1 i i i n m i i =1 j (S , T ) > n (20) > ∑∑ ui Dk Fi (T ) − λi Dk Gi (T ), I Sk − ITk + ∑∑ v j Dk H j (T ), I Sk − I Tk . i =1 k =1 j =1 k =1 Now, by (15) it follows p n ∑∑ u i =1 k =1 i m n Dk Fi (T ) − λi Dk Gi (T ), I Sk − ITk + ∑∑ v j Dk H j (T ), I Sk − I Tk < 0 . j =1 k =1 This inequality contradicts (7). Thus the theorem is proved. Corollary 6. Let S 0 and ( S 0 , u 0 , v 0 , λ 0 ) be feasible solutions to ( Pλ 0 ) and (D), respectively. If the hypotheses of Theorem 5 are satisfied, then S 0 is an efficient solution to ( Pλ 0 ) and ( S 0 , u 0 , v 0 , λ 0 ) is an efficient solution to (D). Proof: We proceed by contradiction. If S 0 is not an efficient solution to ( Pλ 0 ) then there exists a feasible solution S ′ to ( Pλ0 ) such that Fi ( S ′)≤ λi0 Gi ( S ′) , ∀ i ∈ P , and (21) Fj ( S ′) < λ 0j G j ( S ′) , for some j ∈ P . Since ( S 0 , u 0 , v 0 , λ 0 ) is a feasible solution to (D) by (21), and Theorem 5 we obtain a contradiction. Hence S 0 is an efficient solution to (Pλ0 ) . In the same way we obtain that ( S 0 , u 0 , v 0 , λ 0 ) is an efficient solution to (D). Theorem 7. (Strong duality) Let S 0 be a regular efficient solution to (P). Then there exist p u 0 ∈ R +p , ∑ ui0 = 1 , v 0 ∈ R +m , and λ 0 ∈ R +p , such that ( S 0 , u 0 , v 0 , λ 0 ) is a feasible i =1 solution to (D). Further, if the conditions of Weak Duality Theorem 5 also hold, then 70 I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective ( S 0 , u 0 , v 0 , λ 0 ) is an efficient solution to (D) and the values of the objective functions of (P) and (D) are equal at S 0 and ( S 0 , u 0 , v 0 , λ 0 ) respectively. p Proof: Using Theorem 4 we obtain that there exist u 0 ∈ R +p , ∑ ui0 = 1 , v 0 ∈ R +m , and (4) i =1 and (5) hold. Thus, ( S 0 , u 0 , v 0 , λ 0 ) satisfies (7) – (10). Hence, ( S 0 , u 0 , v 0 , λ 0 ) is a feasible solution to (D). Further, if Theorem 5 holds then, by Corollary 6 we obtain that this solution ( S 0 , u 0 , v 0 , λ 0 ) is also an efficient solution to (D), and the values of the objective functions of (P) and (D) are equal at S 0 and ( S 0 , u 0 , v 0 , λ 0 ) respectively. Now we give a strict converse duality theorem of Mangasarian type [19] for (Pλ ) and (D). Theorem 8. (Strict converse duality) Let S * and ( S 0 , u 0 , v 0 , λ 0 ) be efficient solutions to (Pλ0 ) and (D), respectively. Assume that ui0 ( Fi ( S * ) − λi0 Gi ( S * )) * 0 i =1 α i ( S , S ) p p (j1) ∑ ui0 ( Fi ( S 0 ) − λi0 Gi ( S 0 )) ; * , S0) ∑ α (S i =1 i (j2) for any i ∈ P and j ∈ M , ( Fi (⋅) − λi0 Gi (⋅), H j (⋅)) is of d -semistrictly type-I at S * .Then, S 0 = S * . Proof: We assume that S 0 ≠ S * and exhibit a contradiction. Using (j2) we obtain ( Fi ( S * ) − λi0 Gi ( S * )) − ( Fi ( S 0 ) − λi0 Gi ( S 0 )) > n > α i ( S * , S 0 )∑ Dk Fi ( S 0 ) − λi0 Dk Gi ( S 0 ), I S * − I S 0 k k =1 k for any i ∈ P , and n − H j ( S 0 )≥ β j ( S * , S 0 )∑ Dk H j (T ), I S * − I S 0 , j ∈ M . k k =1 k By dividing by α i ( S * , S 0 ) > 0 and β j ( S * , S 0 ) > 0 , respectively, the above inequalities reduce to the following Fi ( S * ) − λi0 Gi ( S * ) Fi ( S 0 ) − λi0 Gi ( S 0 ) − > αi (S * , S 0 ) αi (S * , S 0 ) n ∑ k =1 (22) Dk Fi ( S ) − λ Dk Gi ( S ), I S 0 − I S * 0 0 i 0 k k for any i ∈ P , and − H j (S 0 ) β j (S , S ) * 0 n ∑ k =1 Dk H j ( S 0 ), I S * − I S 0 , j ∈ M k k (23) I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective Multiplying the inequality (22) by u 0 ≥ 0, p ∑u i =1 0 i 71 = 1 , ∀ i ∈ P , and (23) by v 0 ≥ 0, ∀ j ∈ M , and summing after all i and j , respectively , yields p ui0 ui0 * 0 * ( F ( S ) − λ G ( S )) − ( Fi ( S 0 ) − λi0 Gi ( S 0 )) ∑ i i i * * 0 , S0) i =1 α i ( S , S ) p ∑ α (S i =1 i m v 0j H ( S0 ) j =1 β j (S * , S 0 ) −∑ m p n > ∑∑ ui0 Dk Fi ( S 0 ) − λi0 Dk Gi ( S 0 ), I S * − I S * k i =1 k =1 k (24) n + ∑∑ v 0j Dk H j ( S 0 ), I S * − I S 0 . k j =1 k =1 k Now, because ( S 0 , u 0 , v 0 , λ 0 ) is a feasible solution to (D) by (7) we get p p ui0 ui0 * 0 * ( ( ) − λ ( )) − ( Fi ( S 0 ) − λi0 Gi ( S 0 )) − F S G S ∑ i i i * * 0 , S0) i =1 α i ( S , S ) ∑ α (S i =1 i m v 0j H ( S 0 ) i =1 βi ( S * , S 0 ) −∑ (25) > 0. Since v 0j H j ( S 0 ) 0 for any j ∈ M , by (25) we obtain p p ui0 ui0 * 0 * ( F ( S ) − λ G ( S )) > ( Fi ( S 0 ) − λi0 Gi ( S 0 )) ∑ i i i * * 0 ,S0) i =1 α i ( S , S ) ∑ α (S i =1 i which contradicts the assumption (j1). Thus the theorem is proved. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] Bector, C.R., Bhatia, D., and Pandey, S., “Duality for multiobjective fractional programming involving n-set functions”, J. Math. Anal. Appl. 186 (3) (1994) 747-768. Bector, C.R., Bhatia, D., and Pandey, S.,”Duality in nondifferentiable generalized fractional programming involving n-set functions”. Utilitas Math. 45 (1994) 91-96 Begis, D., and Glowinski, R., “Application de la méthode des éléments finis à l’approximation d’une probléme de domaine optimal: Méthodes de résolution de problémes approaches”, Appl. Math. Optim., 2 (2) (1975) 130-169. Bhatia, D., and Kumar, P., “Pseudolinear vector optimization problems containing n-set functions”, Indian J.Pure Appl.Math. 28 (4) (1997) 439-453. Bhatia, D., and Mehra, A., “Lagrange duality in multiobjective fractional programming problems with n-set functions”, J.Math. Anal. Appl. 236 (1999) 300-311. Bhatia, D., and Mehra, A., “Theorem of alternative for a class of quasiconvex n-set functions and its applications to multiobjective fractional programming problems”, Indian J. Pure Appl. Math. 32 (6) (2001) 949-960. Bhatia, D., and Tewari, S., “Multiobjective fractional duality for n-set functions”, J.Inform. Optim. Sci. 14 (3) (1993) 321-334. Cea, J., Gioan, A., and Michel, J., “Quelques résultats sur l’identification de domaines”, Calcolo, 10 (3-4) (1973) 207-232. 72 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] I. M. Stancu-Minasian, G., Dogaru, A., M., Stancu, / Duality for Multiobjective Corley, H.W., “Optimization theory for n-set functions”, J. Math. Anal. Appl., 127 (1) (1987) 193-205. Corley, H.W., and Roberts, S.D., “A partitioning problem with applications in regional design”, Oper. Res., 20(1982) 1010-1019. Dantzig, G., and Wald, A., “On the fundamental lemma of Neyman and Pearson”, Ann. Math. Statistics, 22 (1951) 87-93. Jeyakumar, V., and Mond, B., “On generalised convex mathematical programming”, J. Austral. Math. Soc., Ser. B, 34 (1) (1992) 43-53. Jo, C.L., Kim, D.S., and Lee, G.M., “Duality for multiobjective fractional programming involving n-set functions”, Optimization 29 (3) (1994) 205-213. Kim, D.S., Lee, G.M., and Jo, C.L., “Duality theorems for multiobjective fractional minimization problems involving set functions”, Southeast Asian Bull. Math. 20 (2) (1996) 65-72. Kim, D.S., Jo, C.L., and Lee, G.M., “Optimality and duality for multiobjective fractional programming involving n-set functions”, J. Math. Anal. Appl., 224 (1) (1998) 1-13. Kumar, N., Budharaja, R.K., and Mehra, A., “Approximated efficiency for n-set multiobjective fractional programming”, Asia-Pacific J. Oper. Res. 21 (2) (2004) 197-206. Mishra, S.K., “Duality for multiple objective fractional subset programming with generalized (F, ρ ,σ ,θ ) - V -type-I functions”, J.Global Optim. 36 (4) (2006) 499-516. Mishra, S.K., Wang, S.Y., and Lai, K.K., “Optimality and duality for a multi-objective programming problem involving generalized d -type-I and related n-set functions”. European J.Oper.Res.,173 (2) (2006) 405-418. Mangasarian, O.L., Nonlinear Programming, McGraw-Hill, New York, 1969. Morris, R.J.T., “Optimal constrained selection of a measurable subset”, J. Math. Anal. Appl. 70 (2) (1979) 546-562. Neymann, J., and Pearson, E.S., “On the problem of the most efficient tests of statistical hypotheses”, Philos. Trans. Roy. Soc. London, Ser.A, 231(1933 ) 289-337. Preda, V., “On duality of multiobjective fractional measurable subset selection problems”, J. Math. Anal. Appl.,196 (1995) 514-525. Preda, V., “Duality for multiobjective fractional programming problems involving n-set functions”, in : C.Andreian Cazacu, O.Lehto and Th.M.Rassias (eds.), Analysis and Topology, World Scientific Publishing Company, 1998, 569-583. Preda, V., and Stancu-Minasian, I.M., “Optimality and Wolfe duality for multiobjective programming problems involving n-set functions”. in: N. Hadjisavvas, J. E., Martinez-Legaz, and J. -P., Penot, (eds.), Generalized Convexity and Generalized Monotonocity, Proceedings of the 6th International Symposium on Generalized Convexity/ Monotonocity, Karlovassi, Samos, Greece, 25 Aug.-3 Sep. 1999. Lecture Notes in Economics and Mathematical Systems 502, Springer-Verlag, Berlin, 2001, 349-361. Preda, V., and Stancu-Minasian, I.M., “Mond-Weir duality for multiobjective programming problems involving d-type-I n-set functions”. Rev. Roumaine Math. Pures Appl. 47 (4) (2002) 499-508. Preda, V., Stancu-Minasian, I.M., and Koller, E., “On optimality and duality for multiobjective programming problems involving generalized d-type-I and related n-set functions”, J. Math.Anal.Appl., 283 (1) (2003) 114-128. Stancu-Minasian, I.M., Fractional Programming: Theory, Methods and Applications, Dordrecht, The Netherlands, Kluwer Academic Publishers, pages, 1997, viii + 418. Stancu-Minasian, I.M., and Preda, V., “Optimality conditions and duality for programming problems involving set and n-set functions: a survey”, J.Statist.Manag.Systems, 5 (1-3) (2002) 175-207.