A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem

In this study, we consider the problem of capturing the complete fuzzy non-dominated set of fuzzy multi-criteria optimization problem. We investigate the problem from a fuzzy geometrical viewpoint. The fuzzy feasible regions on the decision space and on the criterion space are formulated using basic fuzzy geometrical ideas. The sup-min composition of the extension principle and the concept of inverse points in fuzzy geometry are used separately to formulate the fuzzy decision feasible region. We show that under a monotone condition on the constraint functions, both of the decision feasible regions are identical. Furthermore, we obtain that the inverse points method can greatly reduce the computational cost in evaluating the fuzzy constraint inequalities. Under the assumption that criteria of the problem are fuzzy number-valued functions, fuzzy points are obtained in the criterion space that correspond to each point in the fuzzy decision feasible region. The complete fuzzy criteria feasible region is the union of all of these fuzzy points. A fuzzy non-dominated set is defined on the formulated fuzzy criteria feasible region. A definition of a proper fuzzy non-dominated point is proposed for making the final decision. Different parts of the proposed methodology are demonstrated with suitable numerical examples and pictorial illustrations.