Z-numbers as generalized probability boxes

Abstract

This paper proposes a new approach to the notion of Z-number, i.e., a pair (A, B) of fuzzy sets modeling a probability-qualified fuzzy statement, proposed by Zadeh. Originally, a Z-number is viewed as the fuzzy set of probability functions stemming from the flexible restriction of the probability of the fuzzy event A by the fuzzy interval B on the probability scale. However, a probability-qualified statement represented by a Z-number fails to come down to the original fuzzy statement when the attached probability is 1. This representation also leads to complex calculations. It is shown that simpler representations can be proposed, that avoid these pitfalls, starting from the remark that when both fuzzy sets A and B 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). Then two proposals are made to generalize this approach when the two sets are fuzzy. One idea is to consider a Z-number as a weighted family of crisp Z-numbers, obtained by independent cuts of the two fuzzy sets, that can be averaged. In the other approach, a Z-number comes down to a pair of possibility distributions on the universe of A forming a generalized p-box. With our proposal, computation with Z-numbers come down to uncertainty propagation with random intervals.