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.
An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality
