The p-Median Problem for Cluster Analysis: A Comparative Test Using the Mixture Model Approach

Recently, Mulvey and Crowder (Mulvey, J., H. Crowder. 1979. Cluster analysis: an application of Lagrangian relaxation. Management Sci. 25 329--340.) suggested that the p-median problem might be useful for cluster analysis problems (where the goal is to group objects described by a vector of characteristics in such a way that objects in the same group are somehow more alike than objects in different groups). The intent of this paper is to test Mulvey and Crowder's proposal using the mixture model approach; i.e., by applying a number of algorithms (including one for the p-median problem) to a set of objects randomly sampled from a number of known multivariate populations and comparing the ability of each algorithm to detect the original populations. In order to evaluate the results, a generalized partition comparison measure and its distribution are developed. Using this measure, results from various algorithms are compared.