Gradual Numbers and Their Application to Fuzzy Interval Analysis

J. Fortin, D. Dubois, H. Fargier
2008 IEEE transactions on fuzzy systems  
We introduce a new way of looking at fuzzy intervals. Instead of considering them as fuzzy sets, we see them as crisp sets of entities we call gradual (real) numbers. They are a gradual extension of real numbers, not of intervals. Such a concept is apparently missing in fuzzy set theory. Gradual numbers basically have the same algebraic properties as real numbers, but they are functions. A fuzzy interval is then viewed as a pair of fuzzy thresholds, which are monotonic gradual real numbers.
more » ... l real numbers. This view enable interval analysis to be directly extended to fuzzy intervals, without resorting to ¢ -cuts, in agreement with Zadeh's extension principle. Several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds. Our approach is illustrated on two known problems: computing fuzzy weighted averages, and determining fuzzy floats and latest starting times in activity network scheduling.
doi:10.1109/tfuzz.2006.890680 fatcat:4ktyltb4qbghjlqkpcwggtpxdi