Conversion of probabilistic information into fuzzy sets for engineering decision analysis

To carry out seismic hazard analysis in the framework of fuzzy set theory, it may become necessary to convert probabilistic information regarding some of the variables into triangular or trapezoidal fuzzy sets. In this paper, three approaches for converting probabilistic information, represented by a probability distribution, into an equivalent triangular or trapezoidal fuzzy set are discussed. In all the three approaches, the probability distribution is first converted into a probabilistic fuzzy set, which is then converted into the equivalent triangular or trapezoidal fuzzy set. The first approach is based on the method of least-square curve fitting, the second approach is based on the conservation of uncertainty (represented by the entropy) associated with the probabilistic fuzzy set in a mean square sense, and the third approach is based on the minimisation of Hausdorff distance (HD) between the probabilistic and the equivalent fuzzy sets. The effectiveness of these approaches in preserving the entropy as well as in preserving the elements of the fuzzy set and their corresponding grades of membership are also discussed with the help of a numerical example of obtaining equivalent fuzzy set for peak ground acceleration. It is found that the approach based on minimisation of Hausdorff distance provides a simple and efficient way for converting the probabilistic information into an equivalent fuzzy set.