Fuzzy optimization problems by parametric representation in the two-dimensional metric space

The authors present a novel representation of fuzzy numbers with bounded supports and also show that a fuzzy number means a bounded continuous curve in the two-dimensional metric space. We consider definitions of the addition, subtraction and product which are defined by the extensions principle due to Zadeh and many other theoreticians of fuzzy logic, as well as showing parametric representation of the operations between fuzzy numbers. The aim of the research is to solve fuzzy optimization problems which consist of nonlinear or linear objective functions and constraints including fuzziness. There are two important problems. One is dealing with some order relation, which is linear or not, and the other is to get a criterion for optimal solutions by considering saddle points of Lagrangian functions.