A note on the admissibility of the centroid-based preorder for fuzzy numbers

Abstract Fuzzy numbers are defined by a membership function, which presents a challenge in comparing and ranking them. The centroid is a commonly used ranking tool as it provides a crisp representative value for a fuzzy number. However, while the centroid induces a total preorder, it lacks antisymmetry, which can lead to indistinguishable rankings for different fuzzy numbers. This limitation has led to the development of admissible orders, which ensure a total order relation that refines the Klir–Yuan partial order. In this paper, we explore the admissibility of the centroid-based preorder. We begin by deriving general formulas for computing the centroid using Riemannian and Lebesguian integrals. Through examples, we show that the centroid does not generally form an admissible preorder, even for fuzzy numbers with continuous membership functions. We then focus on trapezoidal fuzzy numbers and prove that, within this class, the centroid defines an admissible preorder. Finally, we extend this result to interval type 2 fuzzy numbers, demonstrating that the centroid also induces an admissible preorder when the upper and lower membership functions are trapezoidal fuzzy numbers.