International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2013 Vol. 3 (3) July-September, pp.17-24/Dangwal et al. Research Article 17 A GOAL PROGRAMMING PROCEDURE FOR FUZZY MULTI OBJECTIVE LINEAR FRACTIONAL PROBLEM IN VAGUE ENVIRONMENT USING TOLERANCE Rajesh Dangwal 1 , Sharma M.K. 2 and *Padmendra Singh 1 1 Department of Mathematics, H.N.B. Garhwal University, Campus Pauri, Uttarakhand 2 Department of mathematics R.S.S. (P.G) College, Pilkhuwa, Ghaziabad *Author for Correspondence ABSTRACT This paper presents a goal programming (GP) procedure for fuzzy multi objective linear fractional programming (MOLFP) problems under vague environment using tolerance limit. In the proposed approach, which is motivated by Mohamed (Fuzzy Sets and System 89 (1997) 215), GP model for achievement of the highest membership value of each of fuzzy, goals defined for the fractional objectives is formulated. In the solution process, the method of variable change under tolerance limit of the membership and non membership goal associated with the fuzzy goal of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. The approach is illustrated by one numerical example. Key Words: Multiobjective Linear Fractional Programming, Fuzzy Multiobjective Linear Fractional Programming, Fuzzy Programming, Fuzzy Goal Programming, Linear Goal Programming, Membership Function, Non Membership Function, Tolerance Limit INTRODUCTION During the mid-1960s and early 1970s of the last century fractional programming (FP) was studied extensively (Charnes and Cooper, 1962) in the literature. In contrast to the single objective FP, multiobjective fractional programming (MOFP) has not been discussed that extensively and only a few approaches have appeared in the literature (Craven, 1988; Kornbluth and Steuer, 1981) concerning MOFP. It may be pointed out that in most of the MOFP approaches, the problems are converted into single objective FP problems and then solved employing the method of Charnes and Cooper (Charnes and Cooper, 1962). To overcome the computational difficulties of using conventional FP approaches to MOFP problems, the theory of fuzzy sets has been introduced in the field of FP. Linguistic variable approach of Zadeh (Zadeh, 1975) to FMOLFP problem has been proposed by Luhandjula in 1984. Luhandjula‟s approach has been further developed by Dutta et al., (1993) and Dutta (1992). Other approaches in this area have also been investigated (Craven, 1988; Sakawa and Yumine, 1988). In this article, the GP approach to fuzzy programming problems introduced by Mohamed (1997) is extended to solve FMOLFP problems. In the GP model formulation of the problem, first the objectives are transformed into fuzzy goals by means of assigning an aspiration level to each of them. Then achievement of the highest membership value (unity) to the extent possible of each of the fuzzy goals is considered. Present chapter extends the tolerance approach to a special class of fuzzy multi-objective fractional goal programming (FMOFGP) problems in which the fractional objectives essentially have linear terms in the numerator and denominator. In our discussions to follow, we shall refer this class of our approach is that the region of feasible solution in this case is either same or larger than those obtained by other fuzzy goal programming models. This leads to the possibility of arriving at a better solution of problems as Fuzzy Multi-Objective Linear Fractional Goal Programming (FMOLFGP) problems. The Advantage 1. Problem Formulation The general format of a classical multiobjective linear fractional programming problem can be stated as Optimize    =   +      +   , k = 1, 2…., K