International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:02 39 Abstract -- In this paper, the complementary slackness theorem for Seshan’s dual in linear fractional programming problem is proved. A numerical example is presented to demonstrate the result Index Terms -- Duality, Linear Fractional Programming,. I. Introduction N 1960’s and 1970’s several authors, Swarup[11], Bector[1],[2], Chadha [3], Kaska[13], Gol’estein [5],[6], Sharma and Swarup [10], Seshan [9] and many other authors proposed different type of dual problems related to the primal LFP problem consisting in maximizing and minimizing linear – fractional objective function subject to a system of linear constraints. Most of the authors proposed a dual form in which the objective function is linear. Some of them are based on the well known Charnes and Coopers transformation [14] and leads to the duality theory of linear programming. Only Sharma and Swarup [10] , Seshan [9] and Bector [1],[2] have defined a dual form in which the objective function is fractional, that is, ratio of two linear function. Jahan and Islam [12] showed that all these duals proposed by different authors are actually equivalent to one another. Most of the authors proved all the duality theorems. Some of them did not proved Complementary Slackness Theorem .Gol’stein stated this theorem without poof. Also Sharma and Swarup , and Seshan are silent about the Complementary Slackness Theorem. In this paper complementary slackness theorem for Seshan’s dual is proved. II. Dual of a Linear Fractional Program In1972 Swarup and Sharma [10] proposed a dual which has a special feature that both the problem (Primal and dual) are linear fractional. But they considered a primal problem in which constant term does not appear in both the numerator and denominator of the objective function. In 1980 Seshan [9] extended their work to the general case where constant term has permitted to appear in both the numerator and denominator of the objective function and the constraints of the dual are also generalized. Manuscript received March 3, 2010. Sohana Jahan is with the the Department of Mathematics, University of Information Technology and Sciences, Dhaka, Bangladesh (Mobile: +88 - 01918922083, e-mail:sohana_math@yahoo.com) M. Ainul Islam is with the Department of Mathematics, University of Dhaka, Bangladesh. (Mobile: +88 - 01715098891e-mail : mainul_51@yahoo.com) Consider the primal problem (PP) ( PP ) : Maximize β α + + = = x d x c x D x C x f t t ) ( ) ( ) ( (2.1) subject to b Ax ≤ (2.2) 0 ≥ x (2.3) where 0 > + β x d t , S x x x x t n ∈ = ∀ ) ,..., , ( 2 1 , where } 0 , : { ≥ ≤ = x b Ax x S is the feasible set which is assumed to be nonempty and bounded, and that f is not constant on S . A is a m × n matrix, ∈ d c x , , ℜ n ∈ b ℜ m , α , ∈ β ℜ t t d c , denotes transpose of vectors c and d respectively. Seshan [9] proposed the following dual form for the primal problem (PP): (SeDP): Minimize β α + + = u d u c v u g t t ) , ( (2.4) Subject to c d v A u dc u cd t t t β α − ≤ − − 0 ≤ + − v b u c u d t t t β α (2.5) 0 ≥ u , 0 ≥ v , Where ∈ u ℜ n and ∈ v ℜ m This dual form can be written as follows (SeDP): Minimize β α + + = u d u c v u g t t ) , ( Subject to d c u dc u cd v A t t t α β − ≥ + − (2.6) 0 ≥ − − v b u d u c t t t α β (2.7) 0 ≥ u , 0 ≥ v (2.8) A Complementary Slackness Theorem for Linear Fractional Programming Problem Sohana Jahan and M. Ainul Islam I