New inclusion relation of neutrosophic sets with applications and related lattice structure

The main purpose of this paper is to study the inclusion relations of neutrosophic sets and some applications in multiple attribute decision making. At first, the existing two definitions of inclusion relation (called type-1 and type-2 inclusion relation, respectively) of neutrosophic sets are analyzed, and the deficiencies of type-1 and type-2 inclusion relations are illustrated by examples (in fact, they are actually two extreme cases). Second, a new definition of inclusion relation of neutrosophic sets (call it type-3 inclusion relation) is introduced, and a new method of ranking of neutrosophic sets is given. The effectiveness of the ranking method is presented by some application examples in multiple attribute decision making. Finally, type-3 inclusion relation of neutrosophic sets and related lattice structure are investigated in a systematic way, the definitions of type-3 union and type-3 intersection operations are proposed, and the following important result is proved: all of neutrosophic sets based on a certainty domain constitute a generalized De Morgan algebra (non-distributive lattice) with respect to type-3 union, type-3 intersection and complement operations. From this, the essential difference between neutrosophic set and fuzzy set (and intuitionistic fuzzy set) is clarified theoretically.