Multi OBJECTIVE QUADRATIC FRACTIONAL PROGRAMMING (MOQFP) problem involves optimization of several OBJECTIVE functions in the form of a ratio of numerator and denominator functions which involve both contains linear and QUADRATIC forms with the assumption that the set of feasible solutions is a convex polyhedral with a finite number of extreme points and the denominator part of each of the OBJECTIVE functions is non-zero in the constraint set. In this paper, we extend the procedure as suggested by Lachhwani (Proc. Nat. Acad. Sci. India, 82 (4), 317-322) based on fuzzy goal PROGRAMMING approach for the solution of multi OBJECTIVE QUADRATIC FRACTIONAL PROGRAMMING (MOQFP) problem.The proposed technique is simple, efficient and requires less computational work. In the proposed FGP model formulation, corresponding OBJECTIVEs of equivalent multi OBJECTIVE PROGRAMMING problem are transformed into fuzzy goals (membership functions) by means of assigning an aspiration level to each of them and suitable membership function is de fined for each OBJECTIVEs. Then achievement of the highest membership value of each of fuzzy goals is formulated by minimizing the sum of negative deviational variables. The proposed methodology is illustrated with numerical example in order to support the proposed methodology.

In this paper, we establish a proof for a necessary condition for MULTIPLE OBJECTIVE FRACTIONAL pro-gramming. In order to derive the set of necessary conditions, we employ an equivalent parametric problem. Also, we present the related semi parametric model.

In this paper, a multi- OBJECTIVE QUADRATIC PROGRAMMING (Poss- MOQP) problem with possibilistic variables coefficients matrix in the OBJECTIVE functions is studied. Through the use of b-level sets the Poss- MOQP problem is converted into the corresponding deterministic multi- OBJECTIVE QUADRATIC PROGRAMMING (b-MOQP) problem and hence into the single parametric QUADRATIC PROGRAMMING problem using the weighting method. An extended b-possibly efficient solution is specified. A necessary and sufficient condition for finding such a solution is established. A relationship between the solutions of possibilistic levels is constructed. Numerical example is given to clarify the obtained results.

MULTIPLE OBJECTIVE PROGRAMMING (MOP) problems have become famous among many researchers due to more practical and realistic implementations. There have been a lot of methods proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP problems by starting from a utopian point (which is usually infeasible) and moving towards the feasible region via stepwise movements and a plain continuous interaction with Decision Maker (DM). We consider the case where all OBJECTIVE functions and constraints are linear. The implementation of the proposed algorithm is demonstrated with two numerical examples.

By using cross-section data of 1384-85 of North Khorasan Province and MULTIPLE-OBJECTIVE Linear Fuzzy FRACTIONAL PROGRAMMING, this article tries to represent cultivation model corresponding with sustainable agriculture, and compares this model with result of simple Linear PROGRAMMING and present cultivation model. Data were gathered through filling out 627 questionnaires from 55 villages by using two-stage random sampling method. Results of large- scale farms showed that the model offered by linear PROGRAMMING model is near to present cultivation model, meanwhile there is non-negligible difference between sustainable cultivation model and results of linear PROGRAMMING model and present cultivation model. MULTIPLE-OBJECTIVE Linear Fuzzy FRACTIONAL PROGRAMMING will reduce diversification of cultivation model. Based on this PROGRAMMING model, in large-scale farms, irrigated lands cultivation model, will change to wheat and in rain-fed lands, cultivation model will change to wheat and lentil. Also, this optimal cultivation model will reduce Gross Margin of present cultivation model by 51 percentages. This result in short-scale farm is different, to some extent. In order to gain sustainable cultivation model it is not necessary to make change a lot in present cultivation model, in short scale farms irrigated lands cultivation model, will change to wheat, barley and com and rain-fed lands cultivation model will change to wheat, barley and lentil. In this status, this optimal cultivation model will reduce Gross Margin of optimal cultivation model of present cultivation model by 19 percentages.

This paper deals with a multi-OBJECTIVE linear FRACTIONAL PROGRAMMING problem in fuzzy environment. The problem is considered by introducing all the parameters as piecewise QUADRATIC fuzzy numbers. Through the use of the associated real number of the close interval approximation and the order relation of the piecewise QUADRATIC fuzzy numbers, the problem is transformed into the corresponding crisp problem. A proposed method introduces to generate ideals and the set of all fuzzy efficient solutions. The advantage of it helps the decision maker to handle the real life problem. A numerical example is given illustrate the method.