Vania Peneva

Properties of the aggregation operators related with fuzzy relations

Abstract The problem of information generalization in multicriteria decision making is considered in this paper. The information is unified by fuzzy relations and the generalization is realized with the help of aggregation operators. Some of the most oftenly used operators are presented and their properties depending on the properties of the fuzzy relations, which they aggregate, are proved. The sensitivity of the operators with respect to variations in their arguments is investigated. The numerical example deciding the problem of alternatives’ ranking is given as well.

Comparison of clusters from fuzzy numbers

Fuzzy numbers are used for representation of numerical quantities in a vague environment. Their comparison or ranking is important for application purposes. A new index for comparing of fuzzy numbers based on their geometrical properties is suggested in this paper. This geometrical index is tested on a group of selected examples and compared with the other well-known indexes. A method for comparison of m-tuples of fuzzy numbers and an algorithm for comparison of subsets (clusters) of similar, closed m-tuples of trapezoidal fuzzy numbers are presented.

Aggregation of fuzzy preference relations to multicriteria decision making

Weighted aggregation of fuzzy preference relations on the set of alternatives by several criteria in decision-making problems is considered. Pairwise comparisons with respect to importance of the criteria are given in fuzzy preference relation as well. The aggregation procedure uses the composition between each two relations of the alternatives. The membership function of the newly constructed fuzzy preference relation includes t-norms and t-conorms to take into account the relation between the criteria importance. Properties of the composition and new relation, giving a possibility to make a consistent choice or to rank the alternatives, are proved. An illustrative numerical study and comparative examples are presented.

FUZZY LOGIC OPERATORS IN DECISION-MAKING

The fuzzy logic operators are used to decide the following multicriteria decision-making problem. A finite set of alternatives is evaluated by a set of fuzzy criteria, i.e., fuzzy relations, which may be either fuzzy preference, or similarity, or likeness relations. The problem is to construct an evaluation procedure to compare the set of alternatives according to the whole set of criteria, i.e., to aggregate the private fuzzy relations in order to get the union relation as a fuzzy one, enhancing to solve the ranking, choice, or cluster problem. Some properties of these operators, required to decide the ranking, choice, or cluster problems and depending on the properties of the private relations, are proved. A numerical example is given as well.

Fuzzy relations in decision making

Multicriteria decision making problems with an initial information of the type of likeness relations between the couples of alternatives by each criterion are considered. This paper deals with the union of these relations such as the aggregated relation is a likeness relation too. A cluster procedure which combines the similar, likeness alternatives in the subsets (clusters) is proposed on the base of this aggregated relation. A numerical example is given as well.

CLUSTER-a package for cluster analysis

The use of different measures of similarity, clusterization criteria, and theories in cluster analysis makes it difficult to compare interpretations of a single data set. The authors present some methods as well as software tools for comparison of cluster partitions. The package CLUSTER can solve the following basic problems of cluster analysis: (1) How many clusters are present in a given set of objects X if its cluster structure is unknown? (2) If different partitions of X are obtained for a given value of c (the number of clusters), using some clustering methods, then which of these partitions is the most valid? and (3) Are there good clusters for multiple values of c? It can also be used to study the cluster structure of X using different metrics for the calculation of the distance between the objects as well as the dependence of different final partitions on the functionals parameters.<<ETX>>