This paper is directed towards developing two new fuzzy logic-based representations for systems modelling and optimization using the notion of the normalized fuzzy matrices. The first is the arithmetic type based on 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). The second is the visual type based on coloured cells representation expressed by replacing each parameter by its value and a coded (newly defined negative or positive colours based on the colour Hue circle) corresponding to its fuzzy level. For both cases, the theoretical foundations of the fuzzy logic algebra are presented for various cases of operations.