Generalization of arithmetic and visual fuzzy logic-based representations for nonlinear modeling and optimization in fully fuzzy environment

This paper is directed towards the development of the Arithmetic and Visual Logic-based representations for classical nonlinear systems modeling and optimization. The concept was originally proposed by Gabr and Dorrah for linear system using the notion of the normalized fuzzy matrices. In this concept, the arithmetic fuzzy logic-based representation type is suggested based on dual cell representation, expressed by replacing each parameter with a pair of parentheses (Actual Value, Fuzzy Level). The visual fuzzy logic-based type is proposed based on colored cells representation expressed by replacing each parameter by its value and a coded corresponding to its fuzzy level. For both representations, the theoretical foundations of the fuzzy logic algebra, different properties, and analogy with the Conventional Fuzzy Theory are elaborated for various cases of operations. The suggested approach is generalized for the fuzzy case of the classical nonlinear modeling and optimization problems. These problems are normally solved by the Lagrangean Function Method or the Jacobian Technique. The two methods were then modified by incorporating the suggested fuzzy logic-based representations assuming the fuzziness of all the optimization formulation parameters. Using a representative nonlinear optimization numerical example, the proposed fuzzy logic-based formulation is applied. The results demonstrate the consistency and robustness of the developed approach for incorporation with nonlinear optimization problems. Finally, it is shown that the presented concept provides a unified theory for various linear and nonlinear systems´ modeling and optimization in fully fuzzy environments.