Solving a Fully Fuzzy Linear Programming Problem through Compromise Programming

In the current literatures, there are several models of fully fuzzy linear programming (FFLP) problems where all the parameters and variables were fuzzy numbers but the constraints were crisp equality or inequality. In this paper, an FFLP problem with fuzzy equality constraints is discussed, and a method for solving this FFLP problem is also proposed. We first transform the fuzzy equality constraints into the crisp inequality ones using the measure of the similarity, which is interpreted as the feasibility degree of constrains, and then transform the fuzzy objective into two crisp objectives by considering expected value and uncertainty of fuzzy objective. Since the feasibility degree of constrains is in conflict with the optimal value of objective function, we finally construct an auxiliary three-objective linear programming problem, which is solved through a compromise programming approach, to solve the initial FFLP problem. To illustrate the proposed method, two numerical examples are solved.

NORMALIZATION IN TYPE-2 FUZZY RELATIONAL DATA MODEL BASED ON FUZZY FUNCTIONAL DEPENDENCY USING FUZZY FUNCTIONS

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.

Answering Null Queries by Analogical Reasoning on Similarity-based Fuzzy Relational Databases

We present here a method of using analogical reasoning to infer approximate answers for null queries on similarity-based fuzzy relational databases. Null queries are queries that elicit a null answer from a database. Analogical reasoning assumes that if two situations are known to be similar in some respects, it is likely that they will be similar in others. Application of analogical reasoning to infer approximate answers for null queries using fuzzy functional dependency and fuzzy equality relation on possibility-based fuzzy relational database has been studied. However, the problem of inferring approximate answers has not been fully explored on the similarity-based fuzzy relational data model. In this work, we introduce the concept of approximate dependency and define a similarity measure on the similaritybased fuzzy model, as extensions to the fuzzy functional dependency and fuzzy equality relation respectively. Under the framework of reasoning by analogy, our method provides a flexible query answering mechanism for null queries on the similarity-based fuzzy relational data model.