  Maximum diversity problems arise in many practical settings from facility
location to social networks, and constitute an important class of NP-hard
problems in combinatorial optimization. There has been a growing interest in
these problems in recent years, and different mathematical programming models
have been proposed to capture the notion of diversity. They basically consist
of selecting a subset of elements of a given set in such a way that a measure
based on their pairwise distances is maximized to achieve dispersion or
representativeness. In this paper, we perform an exhaustive comparison of four
mathematical models to achieve diversity over the public domain library MDPLIB,
studying the structure of the solutions obtained with each of them. We extend
this library by including new Euclidean instances which permit to analyze the
geometrical distribution of the solutions. Our study concludes which models are
better suited for dispersion and which ones for representativeness in terms of
the structure of their solutions, as well as which instances are difficult to
solve. We also identify in our conclusions one of the models which is not
recommended in any setting. We finalize by proposing two improvements, one
related to the models and one to solving methods. The computational testing
shows the value of the analysis and merit of our proposals.
