Building Transitive Groups in Computer-Supported Collaborative Learning Using Fuzzy Clustering

In cluster analysis context is of major interest to induce an order that allow us to compare the groups generated by cluster analysis itself. Using the sample similarities obtained in this analysis like possibilities in fuzzy sets context provide us to induce a transitivity property in cluster analysis, producing a similarities matrix that meets to the transitivity conditions and that resembles as much as it can to the observed similarities matrix. To find out the transitive similarities matrix it is equivalent to find a solution to a mathematical programming problem where the objective function to be minimized is equal to the absolute difference between the similarities matrix and the unknown transitivity matrix entries. By contrast to non-linear programming analytical or heuristical approaches, which entail tedious and/or intricate calculations, the linear programming model we present in this paper is an easy-computing one that, additionally, provide us the explanation of the associated dual problem, something hardly attainable with other approaches. We include an application relevant in collaborative learning study and the results obtained are commented. Key-words:Cluster analysis, Transitivity property, Similarity matrix, Transitivity matrix, Collaborative learning, Non linear programming