Semi-elliptic membership function: Representation, generation, operations, defuzzification, ranking and its application to the real-time task scheduling problem

A potentially important, yet under-stated membership function (MF) shape viz. semi-elliptic membership function (SEMF) has been elaborated in this paper so that researchers are able to see its true potential for realistic modelling of the decision variables in real-life applications. The concept of semi-elliptic membership function has the practical significance since it can nicely remove the drawbacks associated with the Gaussian membership function shape regarding its long tail with non-zero membership values. A semi-elliptic membership function can be mathematically expressed in terms of only two parameters. It is shown that, with the representation considered here, prominent arithmetic operations such as addition, subtraction, multiplication etc. can be applied on the semi-elliptic fuzzy numbers. As an extension, a novel membership function generation technique for interval type-2 semi-elliptic (IT2SE) MF has also been proposed. Then the applicability of the arithmetic operations on the IT2SE fuzzy numbers is shown. It is illustrated that semi-elliptic fuzzy numbers can suitably be ranked by existing distance measures. We have shown that the well-known INT (Improved Nie-Tan's) defuzzification technique can be applied to the interval type-2 semi-elliptic fuzzy sets for producing defuzzified outputs. Finally, with application point of view, semi-elliptic fuzzy numbers has been applied to the real-time task scheduling problem. Results are compared with triangular fuzzy numbers. Suitable numerical examples are considered for demonstration purposes.