Measuring the Spread of Fuzzy Variable by L-S Integral

In credibility theory, variance is usually used as a measure of the variation of a possibility distribution about the expected value. Since the variance is defined via nonlinear fuzzy integral, its computation is difficult for general fuzzy variables. To avoid this difficulty, this paper defines the spread of a fuzzy variable based on Lebesgue-Stieltjes (L-S) integral. Our approach is to find an "average distance" from the expected value to the possible values of the fuzzy variable. For discrete and several common continuous fuzzy variables such as triangular and normal ones, we present their computation formulas about spread. The relationships between spread and variance are also discussed for common fuzzy variables. The established formulas are useful that can be employed to model practical fuzzy optimization problems.

[1]  Fang-Fang Hao,et al.  Computing the Mean Chance Distributions of Fuzzy Random Variables ∗ , 2008 .

[2]  Ragnar Norberg,et al.  Credibility Theory , 2004 .

[3]  Fang-Fang Hao,et al.  Variance Formulas for Trapezoidal Fuzzy Random Variables , 2009 .

[4]  Yian-Kui Liu,et al.  Convergence of Optimal Solutions about Approximation Scheme for Fuzzy Programming with Minimum-Risk Criteria , 2007, Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007).

[5]  Yian-Kui Liu,et al.  The convergent results about approximating fuzzy random minimum risk problems , 2008, Appl. Math. Comput..

[6]  Baoding Liu,et al.  Uncertainty Theory - A Branch of Mathematics for Modeling Human Uncertainty , 2011, Studies in Computational Intelligence.

[7]  B. van Brunt,et al.  The Lebesgue-Stieltjes Integral , 2000 .

[8]  Yian-Kui Liu,et al.  Expected value of fuzzy variable and fuzzy expected value models , 2002, IEEE Trans. Fuzzy Syst..

[9]  Yian-Kui Liu,et al.  Fuzzy Portfolio Selection Problems Based on Credibility Theory , 2005, ICMLC.

[10]  Fang-Fang Hao,et al.  Mean-variance models for portfolio selection with fuzzy random returns , 2009 .

[11]  Junzo Watada,et al.  Fuzzy random renewal process with queueing applications , 2009, Comput. Math. Appl..

[12]  Jinwu Gao,et al.  The modes of convergence in the approximation of fuzzy random optimization problems , 2009, Soft Comput..

[13]  Arthur P. Dempster,et al.  Classic Works on the Dempster-Shafer Theory of Belief Functions (Studies in Fuzziness and Soft Computing) , 2007 .

[14]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[15]  J. Watada,et al.  T-Independence Condition for Fuzzy Random Vector Based on Continuous Triangular Norms ⁄ , 2008 .

[16]  George J. Klir,et al.  On fuzzy-set interpretation of possibility theory , 1999, Fuzzy Sets Syst..

[17]  Y.-K. Liu,et al.  Convergent results about the use of fuzzy simulation in fuzzy optimization problems , 2006, IEEE Transactions on Fuzzy Systems.

[18]  Yian-Kui Liu,et al.  Modeling fuzzy multi-period production planning and sourcing problem with credibility service levels , 2009, J. Comput. Appl. Math..