Similarity measures for type-2 fuzzy sets and application in MCDM

Type-2 fuzzy set theory is extensively used for decision making, pattern recognition and word computing due to exceptional expression of uncertain information. Similarity measure is one of the core tools for the application of interval and general type-2 fuzzy set. However, there are many similarities exist for interval type-2 fuzzy set, but very few for general type-2 fuzzy set, and the existing similarity measures have some limitations due to their dependence on certain representation, such as one of alpha-plane, zSlices and vertical slice. To solve this problem, in this paper, Dice similarity and Cosine similarity for interval and general type-2 fuzzy set are proposed. The proposed similarity measures for general type-2 fuzzy set are defined on the basis of vector similarity; therefore, they do not depend on certain representation. Furthermore, weighted Dice and Cosine similarity measures are proposed for dealing with special situations in this work. Several properties and a discussion are exposed for proving the presented similarities are indeed similarity measures and can obtain reasonable similarity results. Ultimately, based on presented similarity measures, a multi-criteria decision-making method in the case that criteria weights are completely unknown is proposed. Several examples are employed for illustrating the rationality and superiority of presented similarity measures and MCDM method.