Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship

  • Authors:

  • Wenyi Zeng;Ping Guo

  • Affiliations:

  • College of Information Science and Technology, Beijing Normal University, Beijing 100875, PR China;College of Information Science and Technology, Beijing Normal University, Beijing 100875, PR China

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2008

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Abstract

In this paper, we introduce an axiomatic definition of an interval-valued fuzzy sets' inclusion measure which is different from Bustince's [H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets, Applications to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23 (2000) 137-209]. The relationship among the normalized distance, the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets is investigated in detail. Furthermore, six theorems are proposed showing how the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets can be deduced by the interval-valued fuzzy sets' normalized distance based on their axiomatic definitions. Some formulas have also been put forward to calculate the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets.