A comparison of arithmetical operations with $f$ correlated fuzzy numbers
Diogo Sampaio da Silva, Roberto Antonio Cordeiro Prata
TL;DR
This work develops and analyzes $f$-correlated fuzzy numbers, a generalization of linearly correlated fuzzy numbers defined by a monotone injective function $f$ that links two fuzzy numbers via a joint possibility distribution and the extension principle. The authors derive explicit expressions for the $\alpha$-cuts of both the correlated sum and product, $[A +_f B]^\alpha$ and $[A \cdot_f B]^\alpha$, in terms of the $\alpha$-cuts of $A$ and the function $f$, and study how standard fuzzy arithmetic compares to the correlated case under linear and hyperbolic correlations. They prove that the correlated sum coincides with the standard sum, while the correlated product is contained within the standard product, and they illustrate when equality fails with concrete counterexamples. The paper also demonstrates how certain choices of $f$ (e.g., $f(x)=-x$ or $f(x)=1/x$) yield inverse elements for the correlated sums and products, highlighting structural properties such as invertibility and potential distributivity under specific $f$. Overall, the results provide a coherent framework for computing and comparing arithmetic of interactive fuzzy numbers using only real-number operations and interval endpoints.
Abstract
We present a brief introduction to a class of interactive fuzzy numbers, called $f$-correlated fuzzy numbers, which consist of pairs of fuzzy numbers where one is dependent on the other by a continuous monotone injective function. We have deduced some equations that can directly calculate the results of the sums and products of $f$-correlated fuzzy numbers, using only basic operations with real numbers, intervals on the real line and the function that relates the fuzzy numbers being considered. We proved that their correlated and standard sum coincide, and that in a certain sense, the correlated product is contained in the standard product.