The η-approximation method in granular differentiable fuzzy multiobjective programming

In recent years, there has been a growing interest in modelling real-world extremum problems, processes and systems by optimization problems with uncertain data. One of the approaches for analyzing and solving various classes of mathematical programming problems that are imprecisely defined, because they contain uncertain data, is fuzzy optimization. In this work, we apply the η-approximation method derived by Antczak [ 3 ] in the deterministic smooth case to solve the considered gr -differentiable vector fuzzy optimization problem in which all fuzzy numbers involved are represented by horizontal membership functions. In the aforesaid method used in a fuzzy case, for the original gr -differentiable fuzzy vector optimization problem, the family of its associated vector (deterministic) optimization problems is constructed at the given feasible point. Then, the equivalence between aforesaid multicriteria optimization problems is proved under both granular invexity and classical invexity assumptions. The optimality conditions presented in this paper are also illustrated and analyzed by the appropriate exemplary gr -differentiable vector fuzzy optimization problems. Among them, the exemplary inventory management problem with multiple fuzzy objective mapping is presented which is solved by using the η-approximation method. Finally, the η-approximation procedure is given, which can be applied for solving gr -differentiable fuzzy multiobjective programming problems.