This paper presents further development of the fuzzy logic-based arithmetic representation using the notion of normalized fuzzy logic based representations. The concept was originally proposed by Gabr and Dorrah [1]-[6] for linear and nonlinear system using the notion of the normalized fuzzy matrices. The investigation is based on the dual cell representation, expressed by replacing each parameter with a pair of parentheses, the first is the actual value and the second is corresponding fuzzy level, (Value, Fuzzy Level). Various properties and features of the normalized fuzzy concept are elaborated, including the associative, commutative, distributive and reverses laws. The theoretical foundations of the suggested fuzzy approach logic algebra, different properties, and

The analogy of the proposed Fuzzy Logic-based Arithmetic Representation technique with the Conventional Fuzzy Theory are derived for various cases of operations of addition, subtraction, multiplication and division. It is shown that the suggested approach is identical to that of the conventional fuzzy theory for addition and subtraction operations, and yields similar results of multiplications and divisions operations after ignoring the second order variations terms. Thus, the suggested approach offers additional advantages of linearity, reversibility, simplicity, and applicability. The approach is pragmatic as it requires only specifying heuristically the fuzzy logic-based levels of the parameters and coefficients, which can be relatively evaluated in real life. These levels are then transferred at the end of solution to actual uncertainties.

Application of the proposed fuzzy logic-based representation to a case study of a PERT numerical is presented to demonstrate the efficacy of the proposed technique. Finally, it is demonstrated that for the suggested approach, the equivalent forward path and the backward path of the PERT network are the same due to the linearity property of the concept. Such condition, however, is not satisfied by the Conventional Fuzzy Theory. In addition, for such situations, the solution of the Conventional Fuzzy Theory is very cumbersome and will be completely impractical for high dimension networks.