Spread of fuzzy variable and expectation-spread model for fuzzy portfolio optimization problem

Using Lebesgue-Stieltjes (L-S) integral, we first define the nth moment of a fuzzy variable, and derive some useful moment formulas for common possibility distributions. Particularly, the second moment, called spread, can be represented as quadratic convex functions with respect to fuzzy parameters. Then, we take the spread as a new risk criteria in fuzzy decision systems, combine it with the expectation of fuzzy variable, and develop three classes of fuzzy expectation-spread models for portfolio optimization problems. In the case when the returns are trapezoidal and triangular fuzzy variables, we employ the parametric representations of spreads to turn the proposed expectation-spread models into their equivalent parametric programming problems. In the case when the return parametric matrix is full row rank, the equivalent parametric programming problems become convex programming ones that can be solved by conventional optimization methods or general purpose softwares. Finally, we demonstrate the developed modeling ideas via three numerical examples, and also compare the expectation-spread method with traditional expectation-variance method via a number of numerical experiments. © 2010 Korean Society for Computational and Applied Mathematics.