Using 2-additive fuzzy measure to represent the interaction among if-then rules

Abstract

When fuzzy if-then rules are used to approximate reasoning, interaction exists among rules that have the same consequent. Due to this interaction, the weighted average model frequently used in approximate reasoning may not work well in many real-world problems. In order to handle this interaction, the paper "IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, Volume: 34, No.5, October 2004, pp. 1 -9" proposed to use a non-additive nonnegative set function to replace the weights assigned to rules having the same consequent and to draw the reasoning conclusion based on an integral with respect to the non-additive nonnegative set function. Handling interaction in fuzzy if-then rule reasoning in this way can lead to an improvement of reasoning accuracy. In that paper, the authors proposed an approach to determining the set function when it was not given by the experts. They need to solve a linear programming problem with too many parameters when the number of the rules is large. Actually, it is not feasible to implement in the real world because the number of parameters increases exponentially with the number of rules. This paper proposes a new approach to using the 2-additive fuzzy measure to replace the general set function for handling the interaction among if-then rules. The number of parameters determined in the new approach is greatly less than the number of parameters in the old approach. Compared with the old approach, the new one leads to an accuracy loss to some extent. But the new approach reduces the number of parameters from an exponential to polynomial quantity. It implies that the new approach is feasible and has more wide applications in the real world.