Computational method for fuzzy arithmetic operations on triangular fuzzy numbers by extension principle

Abstract

Fuzzy arithmetic operations are applied to mathematical equations that include fuzzy numbers, which are commonly used to represent non-probabilistic uncertainty in different applications. Although there are two mathematical approaches available in the literature for implementing fuzzy arithmetic (i.e., the α-cut approach, and the extension principle approach), the existing computational methods are mainly focused on implementing the α-cut approach due to its simplicity. However, this approach causes overestimation of uncertainty in the resulting fuzzy numbers, a phenomenon that reduces the interpretability of the results. This overestimation can be reduced by implementing fuzzy arithmetic using the extension principle; however, existing computational methods for implementing the extension principle approach are limited to the use of min and drastic product t-norms. Using the min t-norm produces the same result as the α-cuts and interval calculations approach, and the drastic product t-norm is criticized for producing resulting fuzzy numbers that are highly sensitive to the changes in the input fuzzy numbers. This paper presents original computational methods for implementing fuzzy arithmetic operations on triangular fuzzy numbers using the extension principle approach with product and Lukasiewicz t-norms. These computational methods contribute to the different applications of fuzzy arithmetic; they reduce the overestimation of uncertainty, as compared to the α-cut approach, and they reduce the sensitivity of the resulting fuzzy numbers to changes in the input fuzzy numbers, as compared to the extension principle approach using drastic product t-norm.