Xue-hai Yuan

The cut sets,decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets

In this paper, the cut sets, decomposition theorems and representation theorems of intuitionistic fuzzy sets and interval valued fuzzy sets are researched indail. First, new definitions of four kinds of cut sets on intuitionistic fuzzy sets are introduced, which are generalizations of cut sets on Zadeh fuzzy sets and have the same properties as that of Zadeh fuzzy sets. Second, based on these new cut sets, the decomposition theorems and representation theorems on intuitionistic fuzzy sets are established. Each kind of cut sets corresponds to two kinds of decomposition theorems and representation theorems. Thus eight kinds of decomposition theorems and representation theorems on intuitionistic fuzzy sets are obtained, respectively. At last, new definitions of cut sets on interval valued fuzzy sets are given based on the theory of cut sets on intuitionistic fuzzy sets, and eight kinds of decomposition theorems and representation theorems on interval valued fuzzy sets are also obtained. These results provide a fundamental theory for the research of intuitionistic fuzzy sets and interval valued fuzzy sets.

The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets

The connections among the n-dimensional fuzzy set, Zadeh fuzzy set and the finite-valued fuzzy set are established in this paper. The n-dimensional fuzzy set, a special L-fuzzy set, is first defined. It is pointed out that the n-dimensional fuzzy set is a generalization of the Zadeh fuzzy set, the interval-valued fuzzy set, the intuitionistic fuzzy set, the interval-valued intuitionistic fuzzy set and the three dimensional fuzzy set. Then, the definitions of cut set on n-dimensional fuzzy set and n-dimensional vector level cut set of Zadeh fuzzy set are presented. The cut set of the n-dimensional fuzzy set and n-dimensional vector level set of the Zadeh fuzzy set are both defined as n+1-valued fuzzy sets. It is shown that a cut set defined in this way has the same properties as a normal cut set of the Zadeh fuzzy set. Finally, by the use of these cut sets, decomposition and representation theorems of the n-dimensional fuzzy set and new decomposition and representation theorems of the Zadeh fuzzy set are constructed.

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.

Three new cut sets of fuzzy sets and new theories of fuzzy sets

Three new cut sets are introduced from the view points of neighborhood and Q-neighborhood in fuzzy topology and their properties are discussed. By the use of these cut sets, new decomposition theorems, new representation theorems, new extension principles and new fuzzy linear mappings are obtained. Then inner project of fuzzy relations, generalized extension principle and new composition rule of fuzzy relations are given. In the end, we present axiomatic descriptions for different cut sets and show the three most intrinsic properties for each cut set.

The Category RSC of I-Rough Sets

This paper shall study theory of rough sets from the view of the category theory and topos theory. A category RSC of I-rough sets is presented. Topos properties of the category RSC are studied. It is proved that the category RSC satisfies all properties of a topos except one, for it has no subobject classifier. It is pointed that although the category RSC is not a topos, RSC has a middle object and consequently RSC is a weak topos. By the use of weak topos RSC, the operations of I-rough sets are naturally described.

The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets

Abstract In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of λ -cut sets of IFS and show that λ -cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of λ -cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup.

Setting due dates to minimize the total weighted possibilistic mean value of the weighted earliness-tardiness costs on a single machine

In this paper, it is investigated how to sequence jobs with fuzzy processing times and predict their due dates on a single machine such that the total weighted possibilistic mean value of the weighted earliness-tardiness costs is minimized. First, an optimal polynomial time algorithm is put forward for the scheduling problem when there are no precedence constraints among jobs. Moreover, it is shown that if general precedence constraints are involved, the problem is NP-hard. Then, four reduction rules are proposed to simplify the constraints without changing the optimal schedule. Based on these rules, an optimal polynomial time algorithm is proposed when the precedence constraint is a tree or a collection of trees. Finally, a numerical experiment is given.

Fuzzy hypergroups based on fuzzy relations

Based on fuzzy reasoning in fuzzy logic, this paper studies a fuzzy hyperoperation and a fuzzy hypergroupoid associated with a fuzzy relation. A sufficient and necessary condition for such a fuzzy hypergroupoid being a fuzzy hypergroup is given, and the properties of the fuzzy hypergroups associated with fuzzy relations are investigated. Furthermore, the definition of normal fuzzy hypergroups is put forward and it is shown that the category NFHG of normal fuzzy hypergroups satisfies all the axioms of topos except for the subobject classifier axiom.

The set-valued mapping based on ample fields

Ample fields play an important role in possibility theory. Based on the ample fields in possibility theory, two new concepts such as possibilistic set and possibility mapping are presented. A theoretical approach to define operations and implication operations of fuzzy sets based on the theory of falling shadows of possibilistic sets is established, and the category PosM of ample spaces and possibility mappings and the category @X(@W,A) of possibilistic sets are built up respectively. It is proved that the category @X(@W,A) is a topos.

Cut Sets on Interval-Valued Intuitionistic Fuzzy Sets

The aim of this article is to study the cut sets of interval valued intuitionistic fuzzy sets. By considering the cut sets of an interval-valued intuitionistic fuzzy set as five valued fuzzy sets, the definitions of four cut sets on interval-valued intuitionistic fuzzy sets are given, which are the generalizations of cuts sets on Zadeh fuzzy sets and the intuitionistic fuzzy sets. It is pointed that the cuts sets of interval-valued intuitionistic fuzzy sets have the same properties as that of the intuitionistic fuzzy sets and Zadeh fuzzy sets. These works provide a powerful tool to study interval-valued intuitionistic fuzzy sets.