Duality in the fuzzy-parametric space for fuzzy-parametric nonlinear programming problem

The concepts of fuzziness and parametric analysis are of importance to treat uncertainty in mathematical model and may offer certain more viewpoints. The basic notions of the parametric study in nonlinear programming problem are presented by Osman (Aplikace matematiky 22(5):318–332, 1977 ; Aplikace matematiky 22(5):333–348, 1977 ). In general, a parametric programming problem is not easy to be solved. In addition, sometime, solving a parametric programming problem with parameters in the objective is easier than solving a parametric problem with parameters in the constraints and vice versa. Therefore, a parametric study in duality space is important to facilitate solving a parametric programming problem. The fuzzy nonlinear problem is interested area for research as one of the tools for treating uncertainty. The fuzzy nonlinear problem when parameters in the objective function or constrains or both is called the fuzzy parametric nonlinear problem. Therefore, dealing with fuzziness and duality parametric space is concerned. In this paper, a novelty introduction to the fuzzy basic notions of parametric programming problem are clarified, the relations between the concepts concerning duality in parametric spaces which introduced by Osman et al. (Int J Math Arch 6(12):161–165, 2016 ) and the fuzzy concepts are presented. We present and define the fuzzy parametric notions of the set of feasible parameters, the solvability set, and the stability sets of the first and second kind. These notions are not defined before. The theoretical relations and an illustration example are introduced.