Fuzzy number

About: Fuzzy number is a research topic. Over the lifetime, 35606 publications have been published within this topic receiving 972544 citations.

Neighborhood systems and relational databases

Abstract: Queries in database can be classified roughly into two types: specific targets and fuzzy targets. Many queries are in effect fuzzy targets, however, because of lacking the supports, the user has been emulating them with specific targets by retiring a query repeatedly with minor changes. In this paper, we augment the relational database, with neighborhood systems, so the database can answer a fuzzy query. There have been many works to combine relational databases and fuzzy theory. Bucklles and Petry replaced attributes values by sets of values. Zemankova-Leech, Kandel, and Zviell used fuzzy logic. The formalism of present work is quite general, it allows numerical or nonnumerical measurements of fuzziness in relational databases. The fuzzy theory present here is quite different from the usual theory. Our basic assumption here is that: the data are not fuzzy, the queries are.Motro [Motr86] introduced the notion of distance into the relational databases. From that he can, then, define the notion of “close-ness” and develop goal queries. Though “distance” is a useful concept, yet very often the quantification of it is meaningless or extremely difficult. For example, “very close”, “very far” are meaningful concept of distance, yet there is no practical way to quantity them for all occasions. Our approach here is more direct, we define directly the meaning of “very close neighborhood”. Using the concept of neighborhoods is not very original, in fact, in the theory of topological spaces [Dugu66], mathematician has been using the “neighborhood system” to study the phenomena of “close-ness”. In the territory of fuzzy queries, the notion of “neighborhood” captures the essence of the qualitative information of “close-ness” better than the brute-force-quantified information (distance). A “fuzzy” neighborhood is a qualitative measure of fuzziness.On the surface, it seems a very complicated procedure to define a neighborhood for each value in the attribute. In fact, if we use the characteristic function (membership function) to define a subset, then the defining procedure is merely another type of distance function (non-measure distance or symbolic distance). Now, to define the neighborhood system one can simply re-entered the third column of the relation with linguistic values: “very close”, “close”, “far”. Note that there is a “greater than” relation among these linguistic values. In mathematical terms, they forms a lattice [Jaco60]. For technical reason, we require the values in third column be elements of a lattice. Note that real number is a lattice, so we get Motro's results back.

Extended TODIM method for hybrid multiple attribute decision making problems

Abstract: TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) is a method for solving the multiple attribute decision making (MADM) problem considering decision maker's (DM's) behavior, in which the attribute values are in the format of crisp numbers. It cannot be used to handle hybrid MADM problems with various formats of attribute values. In this paper, an extended TODIM method is proposed to solve the hybrid MADM problem. First, three formats of attribute values (crisp numbers, interval numbers and fuzzy numbers) are expressed in the format of random variables with cumulative distribution functions. Then, according to the concept of the classical TODIM method, the gain and loss matrices concerning each attribute are constructed by calculating the gain and loss of each alternative relative to the others. Further, by calculating the dominance degree of each alternative over the others, the overall value of each alternative can be obtained to rank the alternatives. Finally, two numerical examples are used to illustrate the use of the proposed method.