Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages

We combine the concepts of gradual numbers and Kaucher arithmetic on extended intervals to define extended gradual interval (EGI) arithmetic in which subtraction and division operators are the inverse operators of addition and multiplication, respectively. Use of the proposed EGI operators can lead to non-monotonic gradual intervals that are not fuzzy subsets and cannot be represented by fuzzy intervals. In this context and when fuzzy representation results are desired, an approximation strategy is proposed to determine the nearest fuzzy interval of the non-monotonic gradual interval obtained. This approximation is viewed as an interval regression problem according to an optimization procedure. The EGI operators are applied to the common fuzzy weighted average (FWA) leading to a gradual weighted average (GWA).