An Efficient Computational Framework for Discrete Fuzzy Numbers Based on Total Orders

Discrete fuzzy numbers, and in particular those defined over a finite chain Ln = 0, ldots, n, have been effectively employed to represent linguistic information within the framework of fuzzy systems. Research on total (admissible) orderings of such types of fuzzy subsets, and specifically those belonging to the set D1Lnrightarrow Ym consisting of discrete fuzzy numbers A whose support is a closed subinterval of the finite chain Ln = 0, 1, ldots, n and whose membership values A(x), for x ∈ Ln, belong to the set Ym = 0 = y1 < y2 < cdots < ym-1 < ym = 1 , has facilitated the development of new methods for constructing logical connectives, based on a bijective function, called textitpos function, that determines the position of each A ∈ D1Lnrightarrow Ym. For this reason, in this work we revisit the problem by introducing algorithms that exploit the combinatorial structure of total (admissible) orders to compute the textitpos function and its inverse with exactness. The proposed approach achieves a complexity of O(n2 m log n), which is quadratic in the size of the underlying chain (n) and linear in the number of membership levels (m). The key point is that the dominant factor is m, ensuring scalability with respect to the granularity of membership values. The results demonstrate that this formulation substantially reduces computational cost and enables the efficient implementation of algebraic operations -- such as aggregation and implication -- on the set of discrete fuzzy numbers.