Ordering and ranking fuzzy numbers and their comparisons play a significant role in decision-making problems such as social and economic systems, forecasting, optimization, and risk analysis problems. In this paper, we present a new method for ordering generalized trapezoidal fuzzy numbers with different left and right heights based on the incenter of a triangle with the left and right heights. Using this presented method a new algorithm for ordering fuzzy numbers is given. The proposed method overcomes the certain drawbacks of the existing methods in the literature. The suggested method can order generalized trapezoidal and triangular fuzzy numbers as well as crisp numbers and generalized triangular fuzzy numbers with the same centroid point. We also give an application of the proposed fuzzy ordering method to the fuzzy risk analysis problem for a production system, where the evaluating values are represented by generalized trapezoidal fuzzy numbers with different left and right heights.