Maximum defuzzifier

Maximum Defuzzifier This defuzzifier chooses f as the point at which associated membership functions achieve their maximum values. 

The resulting set must be defuzzified or resolved to a single number (crisp value). Some defuzzification methods are center of area (CoA), bisector, middle of maximum (MOM), largest of maximum, and smallest of maximum (Teti and Kumara 1997). Perhaps the most popular defuzzification method is the center of area (CoA), which returns the center of area under the curve. 

In addition to centroid defuzzifiers, maximum defiizzifiers and means of maxima defuzzifiers are also commonly used. 

Mean of Maxima Defuzzifier This defuzzifier examines fuzzy set B, determines values for which associated membership functions achieve their maximum values and computes the mean of these values as its output j . 

The defuzzified output value has been created by using the MOM (Mean of Maximum) defuzzification method. 

Center-of-Maximum (C-o-M) In the C-o-M method, only the peaks of the membership functions are used. The defuzzified crisp compromise value is determined by finding the place where the weights are balanced. Thus, the areas of the membership functions play no role and only the maxima (singleton memberships) are used. The crisp output is computed as a weighted mean of the term membership maxima, weighted by the inference results. 

The defuzzification process creates a single assessment from the fiizzy cmiclusion set expressing the risk associated with the event, so that corrective actions can be prioritised. Several defuzzification techniques have been developed (Runkler and Glesner (1993)). One conunon technique is the weighted mean of maximum method, which is illustrated here. This technique averages the points of maximum possibility of each fuzzy conclusion, wei ted by their degrees of truth. Hence, if the conclusion fixrm the risk evaluation phase is, for example, 0.5 Low, 0.1 Low and 0.5 Mod, the maximum value for each linguistic term is taken. This reduces the conclusion to 0.5 Low and 0.5 Mod to be defuzzified. 

An inference system commonly used to develop fuzzy models is the Mamdani fuzzy inference system. The Mamdani approach was developed in the 1970s and was the first inference method applied to control systems [15]. The Mamdani inference procedure describes the output variables as fuzzy sets. The approach uses max-min composition in which the minimum of the two antecedents is taken for a particular rule and the maximum combination of the rules is determined for aggregating the effects of aU the rules. The effect of the max combiner on the output membership functions is to generate an "envelope" of the fired output membership functions. In order to defuzzify this output set, the centroid (weighted average) of the envelope is found by integrating over the 2-dimensional shape. The defuzzification process of the Mamdani approach is computationally intensive. 

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