On Approximation of Fuzzy Sets by Crisp Sets: From Continuous Control-Oriented Defuzzification to Discrete Decision Making

In this paper, we show that the necessity to make crisp decisions in uncertain (fuzzy) situations leads to the necessity to "approximate" fuzzy sets by crisp sets. We show that seemingly natural approximation ideas - such as using alpha-cut for a given alpha - often do not work, and we describe new approximations which not only work, but which are optimal in some reasonable sense