Any Improved Trapezoidal Approximation to Intersection (Fusion) of Trapezoidal Fuzzy Numbers Leads to Non-Associativity: A Theorem

In some cases, our uncertainty about about a quantity can be described by an interval of its possible values. If we have two or more pieces of interval information about the same quantity, then we can conclude that the actual value belongs to the intersection of these intervals. In general, we may need a fuzzy number to represent our partial knowledge. A fuzzy number can be viewed as a collection of intervals (α-cuts) corresponding to different degrees α ∈ [0, 1]. In practice, we can only store finitely many α-cuts. Usually, we only store the lower and upper α-cuts (corresponding to α = 0 and α = 1) and use linear interpolation – i.e., use trapezoidal fuzzy numbers. However, the intersection of two trapezoidal fuzzy numbers is, in general, not trapezoidal. One possible approach is to simply take an intersection of lower and alpha α-cuts, but this approach underestimates the resulting membership function. A more accurate approach is to use, e.g., Least Squares Method to provide a better linear approximation to the resulting membership function. However, as we will show, this approximation method makes the corresponding “knowledge fusion” operation non-associative. Moreover, we prove that any improved trapezoidal approximation to intersection (fusion) of trapezoidal fuzzy numbers leads to non-associativity.