The advanced defuzzification methods of the convex α – cut fuzzy sets

The main features of the fuzzy sets and their corresponding membership functions were presented in terms of the fuzzification process and further by the de-fuzzification operation. The convexity of the α -(alfa) cuts of the fuzzy sets is used in the decomposition of the fuzzy sets. The alfa cuts of the fuzzy sets were defined precisely in terms of the pair of functions and their lowest upper and greatest lower bounds. The convex combination of the intervals of the sub-regions of the fuzzy sets and their membership function were considered as the points of the defuzzified values of the fuzzy sets. The methods of the de-fuzzification to the crisp sets were presented by the formulas to find the defuzzification regions and de-fuzzified values. The compositional concepts of the inference as the expansion of the extension principle were introduced to formalize further the fuzzy reasoning by the set of fuzzy rules based on the approximate reasoning.