Representations with fuzzy darts (Q1277862)

The problem raised in this study is concerned with fuzzy darts. Fuzzy darts are developed based on the notion of fuzzy level sets. A fuzzy level set over \({\mathbf X}\) is defined as a structure \(\{(r,A_r) \mid r \in [0,1], A_r\subset {\mathbf X}\}\). That is \((r,A_r)\) associates with an elementary fuzzy set in \({\mathbf X}\) of support \(A_r\) and height ``\(r\)''. A fuzzy number can be represented using level fuzzy sets, namely \([f,g]= \{(r,A_r) \mid\) for \(r\in (0,1)]\}\), meaning that \(A_r= [f(r), g(r)]\) for \(r\in (0,1]\). Subsequently, a fuzzy dart is a fuzzy number \([f,g]\) such that: (i) \(f\) and \(g\) are real valued linear functions on \([0,1]\), (ii) either \(f\) or \(g\) is a constant valued function on \([0,1]\), (iii) \(f\leq g\) on \([0,1]\), (iv) \(f(1)= g(1)\). By partitioning the unit interval into ``\(n\)'' layers of fuzzy strata, fuzzy information can be organized into \(n\)-tuples of fuzzy darts. The main properties of fuzzy darts are revealed. A number of detailed examples are also discussed.