Aggregation of Order-2 Fuzzy Sets

In this article, we are concerned with a problem of aggregation of order-2 information granules, and fuzzy sets, in particular. When processing order-1 fuzzy sets, the structural information about the space over which fuzzy sets are defined is not taken into account at all. In contrast, the aggregation of order-2 fuzzy sets requires a careful attention that needs to be paid both to the closeness determined in the space of membership degrees and the collection of information granules over which such fuzzy sets are defined. We formulate an original optimization problem that simultaneously involves considerations of distances in the membership space (space of membership grades) and some measure of resemblance formed in the space of relationships of reference information granules. The gradient-based learning scheme is constructed. Some illustrative examples are included.