On a multicriteria fuzzy Clustering Method for 3-Way Data

In this paper, we discuss the fuzzy clustering for 3-way data which is composed of objects, attributes and situations. In the case of 2-way data, the clustering of objects are usually based on the values of attributes. However, for 3-way data, a clustering of objects is not always coincident for all situations, because the values of attributes vary with each situation. Therefore, the clustering problem for 3-way data can be regarded as a multicriteria optimization problem. It is known that the practical solutions for a multicriteria optimization problem are Pareto efficient solutions. In a multicriteria hard (non-fuzzy) clustering problem, it is difficult to find a Pareto efficient cluster since this is essentially combinatorial problem. But we show that a multicriteria fuzzy clustering, which is an extension of this hard clustering problem, has merit to obtain Pareto efficient clusters. We investigate the features for Pareto efficient clusters using an artificial 3-way data and we show a numerical example applied to the data of the growth of physical constitutions.