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.

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.