On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering

In this paper we define similarity and inclusion measures between type-2 fuzzy sets. We then discuss their properties and also consider the relationships between them. Several examples are used to present the calculation of these similarity and inclusion measures between type-2 fuzzy sets. We finally combine the proposed similarity measures with Yang and Shih's [M.S. Yang, H.M. Shih, Cluster analysis based on fuzzy relations, Fuzzy Sets and Systems 120 (2001) 197-212] algorithm as a clustering method for type-2 fuzzy data. These clustering results are compared with Hung and Yang's [W.L Hung, M.S. Yang, Similarity measures between type-2 fuzzy sets, International journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004) 827-841] results. According to different alpha-level, these clustering results consist of a better hierarchical tree. (C) 2008 Elsevier Ltd. All rights reserved.