A probabilistic approach to fuzzy and crisp interval ordering

The paper presents a new method of crisp and fuzzy interval comparison (ordering). The method is based on the probabilistic approach and the representation of fuzzy numbers as ordered α-level sets. It allows all the cases of interval location and overlapping to be taken into account, including the ordering of intervals and real numbers. Additionally, the method implicitly allows the widths of intervals to be used in ordering procedures. It should be noted that the probabilistic approach was employed only to infer the set of formulas needed to estimate quantitatively the degree to which one interval is less than or equal to another interval. However, the measure of this value may be treated as probability. Some simple examples are also presented to illustrate the technique’s practical efficiency.