'Useful' fuzzy measures of information, integrated ambiguity and directed divergence

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In the present paper, a new concept of 'useful' fuzzy information, based on utility, is introduced by considering the uncertainties of fuzziness and probabilities of random events. A 'useful' fuzzy measure of integrated ambiguity is obtained by integration of fuzzy and probabilistic uncertainties with utility. A new 'useful' measure of fuzzy-directed divergence of a fuzzy set from another fuzzy set is proposed and its validity proved. Finally, the constrained optimisation of 'useful' fuzzy entropy and 'useful' fuzzy-directed divergence is studied. 1. Introduction Fuzzy set theory (Zadeh 1965) makes use of entropy to measure the degree of fuzziness in a fuzzy set, which is also called fuzzy entropy (Ebanks 1983, Pal and Bezdek 1994). Fuzzy entropy is the measurement of fuzziness in a fuzzy set, and thus has an important position in fuzzy systems, such as fuzzy pattern recognition systems, fuzzy neural network systems, fuzzy knowledge base systems, fuzzy decision making systems, fuzzy control systems and fuzzy management information systems. Let S be any set. A fuzzy subset A in S is characterised by a membership function m A : S ! ½0; 1Š. The value m A ðxÞ represents the grade of membership of x [ S in A. The membership function may be described as follows: