Granular representation and granular computing with fuzzy sets

  • Authors:

  • Adam Pedrycz;Kaoru Hirota;Witold Pedrycz;Fangyan Dong

  • Affiliations:

  • Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama-city 226-8 ...;Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama-city 226-8 ...;Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6R 2V4 and Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz ...;Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama-city 226-8 ...

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2012

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Abstract

In this study, we introduce a concept of a granular representation of numeric membership functions of fuzzy sets, which offers a synthetic and qualitative view at fuzzy sets and their ensuing processing. The notion of consistency of the granular representation is formed, which helps regard the problem as a certain optimization task. More specifically, the consistency is referred to a certain operation @f, which gives rise to the concept of @f-consistency. Likewise introduced is a concept of granular consistency with regard to a collection of several operations, Given the essential role played by logic operators in computing with fuzzy sets, detailed investigations include and- and or-consistency as well as (and, or)-consistency of granular representations of membership functions with the logic operators implemented in the form of various t-norms and t-conorms. The optimization framework supporting the realization of the @f-consistent optimization process is provided through particle swarm optimization. Further conceptual and representation issues impacted processing fuzzy sets are discussed as well.