cvar reduced fuzzy variables and their second order moments

based on credibilistic value-at-risk (cvar) of regularfuzzy variable, we introduce a new cvar reduction method fortype-2 fuzzy variables. the reduced fuzzy variables arecharacterized by parametric possibility distributions. we establishsome useful analytical expressions for mean values and secondorder moments of common reduced fuzzy variables. the convex properties of second order moments with respect to parameters are also discussed. finally, we take second order moment as a new risk measure, and develop a mean-moment model to optimize fuzzy portfolio selection problems. according to the analytical formulas of second order moments, the mean-moment optimization model is equivalent to parametricquadratic convex programming problems, which can be solved by general-purpose optimization software. the solution results reported in the numerical experiments demonstrate the credibility of the proposed optimization method.