Midpoints as average representations of pairs of descriptions by means of fuzzy subsets

Let d be a metric on the set FP(X) of fuzzy subsets of a set X. A midpoint of two fuzzy subsets μ, ν ∈ FP(X) is any fuzzy subset ξ ∈ FP(X) such that d(ξ, μ) = d(ξ, ν) = 1 2d(μ, ν). These midpoints can be used to describe “middle ways” or “compromises” between two situations described by the fuzzy subsets μ and ν. In this work we explicitly compute midpoints for weighted Hamming distances and for weighted maximum distances. The former is a generalization of a previous work by Nieto and Torres (Artif. Intell. Med. 27 (2003), 81-101). We also propose a new application of midpoints in medicine, based on their use as average representations of patients of which we have available two descriptions as fuzzy subsets of a set of attribute variables.