Weighted semi-trapezoidal approximations of fuzzy numbers

  • Authors:

  • Chi-Tsuen Yeh

  • Affiliations:

  • Department of Applied Mathematics, National University of Tainan, 33, Sec. 2, Shu-Lin St., Tainan 70005, Taiwan

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2011

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Abstract

Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These publications can be grouped into two classes: Euclidean distance class and non-Euclidean distance class. Most approximations in Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. In the present paper, we study a special class of non-linear approximations with respect to a weighted Euclidean distance. We call it ''weighted semi-trapezoidal approximations''. The proposed approximations generalize all recent approximations in Euclidean distance class. First, we embed fuzzy numbers into a Hilbert space. Then compute weighted semi-trapezoidal approximations by means of best approximations in a closed convex subset of the Hilbert space. Finally, we propose formulas of matrix type, which is more clear than the previous contributions.