Centroids Beyond Defuzzification

In general, expert rules expressed by imprecise (fuzzy) words of natural language like "small" lead to imprecise (fuzzy) control recommendations. If we want to design an automatic controller, we need, based on these fuzzy recommendations, to generate a single control value. A procedure for such generation is known as defuzzification. The most widely used defuzzification procedure is centroid defuzzification, in which, as the desired control value, we use one of the coordinates of the center of mass ("centroid") of an appropriate 2-D set. A natural question is: what is the meaning of the second coordinate of this center of mass? In this paper, we show that this second coordinate describes the overall measure of fuzziness of the resulting recommendation.