Fuzzy Monte Carlo Simulation using point-cloud-based probability–possibility transformation

Fuzzy Monte Carlo Simulation (FMCS) uses both the probability density function (pdf) and possibility distributions (e.g., fuzzy numbers) to model the uncertainty/imprecision associated with the input parameters and, then, to simulate the uncertainty/imprecision associated with the output parameters. A probability–possibility transformation is needed to transfer the information of a fuzzy number into its equivalent pdf, while performing the simulation. This study deals with an approach of FMCS that uses a point-cloud-based probability–possibility transformation. Let x(t), t = 0,1,…, n, be a set of points that represents some random states of an uncertain/imprecise quantity. The collection of points (x(t), x(t+i)), t = 0,…, n–i, i∈ {1,2,…} is called point-cloud, providing a visual/computational representation of variability, modality, and ranges associated with the quantity. This study identifies the pdf and possibility distribution (fuzzy number) underlying a given point-cloud. Using these distributions, the relationships between the triangular fuzzy number and unimodal pdf (normal/uniform distributions) are identified. Two numerical examples are described elucidating the effectiveness of the proposed transformation. The first example deals with the issue of monitoring a FMCS process. The other example deals with the issue of making a decision by using FMCS.