Fuzzy modeling Mamdani models

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. 

The described inference method, due to Mamdani (1974), is the most popular other inference methods for fuzzy systems based on linguistic rules are Sugeno models and Tsukamoto models (Mamdani 1974 Sugeno 1985). 

Inference Engine Inference Engine maps input type-2 fuzzy sets into output type-2 fuzzy sets by applying the consequent part where this process of mapping from the antecedent part into the consequent part is interpreted as a type-2 fuzzy implication which needs computations of union and intersection of type-2 fuzzy sets. The inference engine in Mamdani system maps the input fuzzy sets into the output fuzzy sets then the defuzzifier converts them to crisp outputs. The rules in Mamdani model have fuzzy sets in both the antecedent part and the consequent part. For example, the /th rule in a Mamdani rule base can be described as follows ... 

Tavanai et al (2005) developed a fuzzy system to model the color yield k/s) of polyester (PET) yarns as a function of time, temperature, and dispersed dye concentration. A typical rule for the color yield was if (temperature is low) and (time is low) and (concentration is low), then (color yield is very low). Only a small number of rules were necessary to create a viable model. In a later paper, Nasiri et al (2011) developed this model by further offering a compromise between the accuracy of classic Takagi-Sugeno systems and the interpretability of Mamdani models by applying genetic algorithms to improve the fuzzy rules. 

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