Fuzzy Lagrange interpolation method from summation of interactive fuzzy numbers

Abstract

This article proposes a novel approach to extending the sum of interactive fuzzy numbers, which is independent of the order of its operands. Interactive fuzzy numbers are fuzzy quantities in which the values across different $\alpha$-cuts are not assumed to vary independently, incorporating dependencies that better reflect real-world uncertainty. A characterization of the proposed summation is given in terms of $\alpha$-cuts, making computational implementation easier. It is shown that this operation preserves essential mathematical properties, including associativity. This is particularly important, as it enables the consistent aggregation of multiple fuzzy quantities without concern for the order in which the operands are grouped. The norm and width behaviors under this new summation are also analyzed. To illustrate the theoretical results, several examples are provided. As a practical application, the classical Lagrange polynomial interpolation method is extended to handle uncertain parameters represented by interactive fuzzy numbers. A fuzzy curve fitting problem is examined using this framework, and a comparative discussion highlights the advantages of the proposed method over existing approaches.