International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

A new class of fuzzy implications, called (g,min)-implications, is introduced by means of the additive generators of continuous Archimedean t-conorms, called g-generators. Basic properties of these implications are discussed. It is shown that the (g,min)-implications are really a new class different from the known (S,N)-, R-, QL- and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz. law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are proposed, under which the corresponding functional equations are satisfied.

Given a Brouwerian complete lattice (L,≤) and two referential sets K and E, and using a fuzzy relation R ∈ L^E×E which is reflexive and symmetric, certain fuzzy relations are characterized as solutions of X ⊲ R = X, proving that they play the role of the blocks in the context of crisp tolerance relations. Moreover, it is verified that these new fuzzy blocks can be determinated by means of the L-fuzzy concepts associated with the K-labeled L-fuzzy context (L,K,E,E,R).

In a previous paper¹, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open.

The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets.

Group decision making is an important category of problem solving techniques for complicated problems, among which the Delphi method has been widely applied. In this paper an improved Delphi method based on Cloud model is proposed in order to deal with the fuzziness and uncertainty in experts' subjective judgments. The proposed Cloud Delphi Method (CDM) describes experts' opinions by Cloud model and we aggregate the experts' Cloud opinions by synthetic algorithm and weighted average algorithm. Another key point of CDM is to stabilize and accommodate the individual fuzzy estimates by the defined stability rules rather than having to force them to converge, or reduce. The Cloud opinions and aggregation results can be exhibited in a graphically way leading experts to judge intuitively and it can decrease the number of repetitive surveys and/or interviews. Moreover, it is more scientific and easier to represent experts' opinion base on Cloud model which can combine fuzziness and uncertainty well. A numerical example is examined to demonstrate applicability and implementation process of CDM.

In fuzzy relational databases, the data dependencies, especially the fuzzy functional dependency(ffd) plays an important role in maintaining the consistency of the database and in avoiding the redundant storage of the data. In the past, it has been shown that the type-2 fuzzy relational databases captures impreciseness and incompleteness in data in a better way. The aim of this paper is to provide the concepts for database normalization in a type-2 fuzzy relational database, so that the normalized schemas can be obtained. Here, we deal with the fuzzy functional dependency(ffd) based normalization of type-2 fuzzy relational databases. We use the concepts of fuzzy functions to derive the fuzzy equality and using this fuzzy equality, we define a new definition of fuzzy functional dependency. First we discuss various approaches proposed by the researchers in this context and show why our fuzzy functional dependency is better, as compared to the earlier ffds proposed by the researchers. We call our ffd as non-0 LHS ffd. We identify an anomaly called "spurious ffd" and show that some of the significant contributions proposed by the earlier researchers are suffering from this anomaly, but the non-0 LHS ffd does not suffer from it. Then, we prove that the set of inference rules for the non-0 LHS ffd are sound and complete. We use the definition of non-0 LHS ffd in obtaining the first three normal forms upto BCNF for type-1 and type-2 fuzzy relational schemas. The result of the decomposition and the procedure to obtain the membership value of the decomposed relations is proposed. The associated concepts like the fuzzy key, fuzzy superkey, fuzzy foreign key are defined in terms of non-0 LHS ffd. On the basis of these concepts, we define full ffd, partial ffd etc. In the last, we show that in our case, the relationship of total-ordering between the three normal forms in classical relational databases is also observed.

The aim of this paper is to develop some new linguistic aggregation operators, such as linguistic power geometric (LPG) operator, linguistic weighted PG operator, LPOWG operator which are based on PG operator. We have studied some desired properties of the developed operators, such as commutativity, idempotency, boundary, etc. Moreover, we have developed two approaches to deal with group decision making problems under multiplicative linguistic preference relations. If the weighting vector of the decision makers is known, we develop an approach which is based on the linguistic weighted PG operator. On the other hand, if the weighting vector of the decision makers is unknown, we develop another approach which is based on the LPOWG. Finally, a practical example is given to illustrate the multiple attribute group decision making process; a comparative study to the linguistic weighted geometric average (LWGA) operator method is also demonstrated.