The Three-dimensional Fuzzy Sets

A new type of L—fuzzy set called 3-dimensional fuzzy set is introduced in this paper. It extends Zadeh fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets. We consider the 3-dimensional fuzzy sets with their basical theories which include concepts of cut sets, decomposition theorems, representation theorems and extension principles. We construct categories of 3-dimensional fuzzy sets, present a membership-relation between a fuzzy point and a 3-dimensional fuzzy sets, and study decision making problem based on the 3-dimensional fuzzy sets. More specially we present the results as follows.1. In section 2, we consider the 3-dimensional fuzzy sets with their basical theories. First, we give definitions of 4 types of cut sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as those of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems, based on the cut sets of 3-dimensional fuzzy sets and 4-valued nested sets. Therefore we establish relations between 3-dimensional fuzzy sets and 4-valued fuzzy sets. Last, we construct category QFuz of 4-valued fuzzy sets , category TPuz of 3-dimensional fuzzy sets and category QNS of 4-valued nested sets respectively. We prove that categories QFuz and TFuz are weak toposes while QNS is a topos. Therefore we conclude that the category of 3-dimensional fuzzy sets has the same topos properties as that of Zadeh fuzzy sets.2. In section 3, we consider the relations between 4-valued fuzzy sets and Zadeh fuzzy sets. First, we put forward 3-dimensional vector level cut sets of Zadeh fuzzy sets which are 4-valued fuzzy sets. The cut sets defined in this way have the same properties as A—cut sets of Zadeh fuzzy sets. Second, we develop 8 decomposition theorems and 8 representation theorems based on the 3-dimensional vector level cut sets. Therefore we establish relations between Zadeh fuzzy sets and 4-valued fuzzy sets.3. In section 4, we first introduce the 3-dimensional fuzzy relations, and describe their composition, inner-composition, projection, and inner-projection by the representation theorems. We then develop the extension principles of the 3-dimensional fuzzy sets which include the maximum extension principle, the minimum extension principle, the maximum multiple extension principle, the minimum multiple extension principle and generalized extension principle. Last we put forward two kinds of fuzzy linear transformation of 3-dimensional fuzzy sets, and present two composition methods of a 3-dimensional fuzzy relation and a 3-dimensional fuzzy set.4. In section 5, we study convex 3-dimensional fuzzy sets and fuzzy decision making problem based on 3-dimensional fuzzy sets. First, we give definition of membership-relation between a fuzzy point and a 3-dimensional fuzzy set, and present definition of (α,β)—convex 3-dimensional fuzzy sets. we show that, in 16 kinds of (α,β)—convex 3-dimensional fuzzy sets, the significant are (∈,,∈)—,(∈,∈∨q)—and ((∈|-), (∈|-)∨(q|-))—convex 3-dimensional fuzzy sets. Furthermore, we give definitions of convex 3-dimensional fuzzy sets with thresholds and describe them using the membership-relation between a fuzzy point and a 3-dimensional fuzzy set. Second, we construct a TOPOSIS method on fuzzy decision making problem based on 3-dimensional fuzzy sets. Last, by introducing cut functions based on 3-dimensional vectors, we give the definition of comparision possibility degree that compares one 3-dimensional vector with another and its expressions for 20 cases. Relating to these, we develop a fuzzy decision making method based on 3-dimensional vectors.