Chapter 4: Fuzzy Numbers and Arithmetic
4.1 Introduction to Fuzzy Numbers In classical mathematics, numbers are precise and well-defined. However, real-world situations often involve uncertainty, vagueness, or imprecision. Fuzzy numbers are an extension of real numbers that incorporate such uncertainty. Definition of Fuzzy Numbers A fuzzy number is a fuzzy set defined on the real number line R , which is: Normal : At least one element has a membership value of 1. Convex : For any two real numbers x and y in the fuzzy set and for all λ in [0, 1], μ(λx + (1 - λ)y) ≥ min(μ(x), μ(y)) . Upper semi-continuous . Support is bounded : The set of elements with non-zero membership grades is a bounded interval. Common Types of Fuzzy Numbers Triangular Fuzzy Number (TFN) : Defined by a triplet (l, m, u) where: l = lower limit m = modal (peak) value with membership 1 u = upper limit The membership function: Trapezoidal Fuzzy Number (TrFN) : Defined by four points (a, b, c, d) : ...