ϕ-calculus: A new fuzzy arithmetic

This article proposes a new arithmetic for fuzzy numbers called ϕ-calculus developed at the LAMIH laboratory (D. Roger and J.-M. Lecomte, 1996). For this algebra, the modeling used for the representation of fuzzy numbers is the distribution function instead of the classical membership function. New functions (square, square root, power, exponential, Neperian logarithm and trigonometric functions) are defined in this paper. With the aim of handling simply ϕ-calculus arithmetic and of comparing it with those of the intervals theory (R.E. Moore, 1966) and the Zadeh´s extension principle for fuzzy numbers (L.A. Zadeh, 1965), a toolbox for Matlab has been set up. Also, several examples and an application of automatic control illustrate this algebra.