On the centroids of fuzzy numbers

In a paper by Cheng [A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95 (1998) 307–317], a centroid-based distance method was suggested for ranking fuzzy numbers, both normal and non-normal, where the fuzzy numbers are compared and ranked in terms of their Euclidean distances from their centroid points to the origin. It is found that the centroid formulae provided by the above paper are incorrect and have led to some misapplications. In this paper we present the correct centroid formulae for fuzzy numbers and justify them from the viewpoint of analytical geometry. A numerical example demonstrates that Cheng's formulae can significantly alter the result of the ranking procedure.