‘Multidimensional’ extensions and a nested dual approach for the m-median problem

A general framework for modeling median type locational decisions, where (i) travel costs and demands may be stochastic, (ii) multiple services or commodities need to be considered, and/or (iii) multiple median type objectives might exist, is presented—using the concept of “multidimensional networks”. The classical m-median problem, the stochastic m-median problem, the multicommodity m-median problem and and multiobjective m-median problem are defined within this framework.By an appropriate transformation of variables, the multidimensional m-median problem simplifies to the classical m-median problem but with a K-fold increase in the number of nodes, where K is the number of dimensions of the network. A nested dual approach to solve the resulting classical m-median problem, that uses Erlenkotter's facility location scheme as a subroutine, is presented. Computational results indicate that the procedure may perhaps be the best available one to solve the m-median problem exactly.