Selection of fuzzy membership function based on probabilistic confidence

The key element of fuzzy set theory is the selection of membership function with suitable confidence. The variation of membership function of a fuzzy variable leads to uncertainty of the estimated model parameters. If a fuzzy variable follows certain random distribution then the fuzzy variable can be called as a fuzzy random variable. The representation of fuzzy random variable with suitable membership function requires mathematical operations such as identification of fuzzy component and random component of the fuzzy-random variable, definition of fuzzy component and random component, application of vertex theory to get all combination of fuzzy intervals for a specified degree of impreciseness of fuzzy component, estimation of cumulative density function for maximum and minimum among all combination of fuzzy intervals, and finally determination of membership function based on specified percentile value from the cumulative density function. All these aspect for selection of membership function of a fuzzy random variable have been emphasized in this paper. A numerical case study has also been demonstrated on fuzzy random variable i.e. change of temperature, commonly used in temperature controller. The variation of uncertainty with types of membership function, with percentile value and with alpha-cut value has been estimated.