Development of fuzzy logic-based arithmetic and visual representations for solving quadratic programming in fully fuzzy environment

This paper presents the development of fuzzy logic representations using the notion of normalized fuzzy matrices developed by Gabr and Dorrah [1–4] for solving quadratic programming problems in a fully fuzzy environment. 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 colored cells representation expressed by replacing each parameter by its value and coded (negative or positive) colors based on the color Hue circle corresponding to its fuzzy level. The quadratic programming problem formulation in its general form is developed in a fully fuzzy environment. A modified dual simplex method algorithm is depicted for the representation of the equivalent linear optimization problem. The problem is represented in a spreadsheet model with built-in programmed Visual Basic Applications macros. The proposed fuzzy logic algebra is then used in a straightforward manner inside this spreadsheet model. The fuzzy logic levels can be easily transferred at the end of the solution to equivalent uncertainties (each level is substituted by a corresponding actual mean and actual standard deviation). Finally, a numerical example is given to illustrate the efficacy of the developed formulations.