An entropy framework for randomness and fuzziness

In the real world, it is common to observe both randomness and fuzziness concurrently. Accordingly, an event/system/decision has multiple possibilities. Thus, in order to compare multiple uncertain options, a measure of overall uncertainty is often desired. This is the topic of this paper. A versatile and intuitive framework that could quantify concurrent probabilistic, fuzzy, and prob-fuzzy uncertainties is proposed. Based on the same, new entropy functions are developed to quantify the prob-fuzzy uncertainty. The proposed entropy functions are inspired from the popular entropy functions such as Shannon's entropy. The properties of the entropy functions are rigorously analysed. A real case-study in the agriculture and environment domain is included to demonstrate the usefulness of the work.