tailieunhanh - Duality for multiobjective fractional programming problems involving d-type-I n-set functions

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. | Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 63-73 DOI: DUALITY FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS INVOLVING d -TYPE-I -SET n - FUNCTIONS 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 .