A Fresh Look at Z-numbers – Relationships with Belief Functions and p-boxes

ABSTRACT This paper proposes a new approach to the notion of Z-number, i.e., a pair of fuzzy sets modelling a probability-qualified fuzzy statement. Originally, a Z-number is viewed as the fuzzy set of probability functions stemming from the flexible restriction of the probability of a fuzzy event by a fuzzy probability. This representation leads to complex calculations and does not reduce to the original fuzzy event when the attached probability is 1. Simpler representations are proposed, that avoid these pitfalls. We note that when both fuzzy sets forming the Z-number are crisp, the generated set of probabilities is representable by a special kind of belief function that corresponds to a probability box (p-box). Two proposals are made to generalise this approach when the two sets are fuzzy. One approach considers a Z-number as a weighted family of crisp Z-numbers, obtained by independent cuts of the two fuzzy sets. In the alternative approach, a Z-number can be turned into a pair of possibility distributions forming a generalized p-box. In that case, the probability of each cut of the fuzzy event is upper and lower bounded by two probability values. Then computation with Z-numbers come down to uncertainty propagation with random intervals.