On Parametric Multi-level Multi-objective Fractional Programming Problems with Fuzziness in the Constraints
M. S. Osman
Department of Mathematics and Operations Research, El Asher University, Egypt.
O. E. Emam
Department of Information Systems, Faculty of Computers and Information, Helwan University, P.O.Box 11795, Egypt.
M. A. El Sayed *
Department of Basic Engineering Sciences, Faculty of Engineering, Benha University, El Qalyoubia, Egypt.
*Author to whom correspondence should be addressed.
Abstract
In this paper, a new concept of the fuzzy stability set of the first kind for multi-level multi-objective fractional programming (ML-MOFP) problems having a single-scalar parameter in the objective functions and fuzziness in the right-hand side of the constraints has been introduced. Firstly, A parametric ML-MOFP model with crisp set of constraints is established based on the α-cut approach. Secondly, a fuzzy goal programming (FGP) approach is used to find an α-Pareto optimal solution of the parametric ML-MOFP problem. Thus, the FGP approach is used to achieve the highest degree of each membership goal by minimizing the sum of the negative deviational variables. Finally, the fuzzy stability set of the first kind corresponding to the obtained α-Pareto optimal solution is developed here, by extending the Karush-Kuhn-Tucker optimality conditions of multi-objective programming problems. An algorithm to clarify the developed fuzzy stability set of the first for parametric ML-MOFP problem as well as Illustrative numerical example are presented.
Keywords: Multi-level programming, multi-objective programming, fractional programming, fuzzy sets, parametric programming