Fuzzy sets and decision theory

The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the “entropy” of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the “energy” and the “effective power” of a fuzzy set, defined as ∑i=1Nwifiandφ∑i=1Nfi, respectively, where w is a nonnegative weight function and φ a nonnegative constant. If w = constant and φ ≠ 0, the energy is proportional to the effective power and, therefore, to the “power” of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed