Accuracy and complexity evaluation of defuzzification strategies for the discretised interval type-2 fuzzy set.

Abstract

The work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised interval type-2 fuzzy sets. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this bottleneck. In this paper we survey four alternative strategies for defuzzifying an interval type-2 fuzzy set: 1. The Karnik-Mendel Iterative Procedure, 2. the Wu-Mendel Approximation, 3. the Greenfield-Chiclana Collapsing Defuzzifier, and 4. the Nie-Tan Method. We evaluated the different methods experimentally for accuracy, by means of a comparative study using six representative test sets with varied characteristics, using the exhaustive method as the standard. A preliminary ranking of the methods was achieved using a multi-criteria decision making methodology based on the assignment of weights according to performance. The ranking produced, in order of decreasing accuracy, is 1. the Collapsing Defuzzifier, 2. the Nie-Tan Method, 3. the Karnik-Mendel Iterative Procedure, and 4. the Wu-Mendel Approximation. Following that, a more rigorous analysis was undertaken by means of the Wilcoxon Nonparametric Test, in order to validate the preliminary test conclusions. It was found that there was no evidence of a significant difference between the accuracy of the Collapsing and Nie-Tan Methods, and between that of the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. However, there was evidence to suggest that the collapsing and Nie-Tan Methods are more accurate than the Karnik-Mendel Iterative Procedure and the Wu-Mendel Approximation. In relation to efficiency, each method’s computational complexity was analysed, resulting in a ranking (from least computationally complex to most computationally complex) as follows: 1. the Nie-Tan Method, 2. the Karnik-Mendel Iterative Procedure (lowest complexity possible), 3. the Greenfield-Chiclana Collapsing Defuzzifier, 4. the Karnik-Mendel Iterative Procedure (highest complexity possible), and 5. the Wu-Mendel Approximation.