Properties of Distance Functions and Minisum Location Models

Abstract

This study is divided into two main parts. The first section deals with mathematical properties of distance functions. The fp norm is analyzed as a function of its parameter p, leading to useful insights for fitting this distance measure to a transportation network. Properties of round norms are derived, which allow us later to generalize some well-known results. The properties of a norm raised to a power are also investigated, and these prove useful in our subsequent analysis of location problems with economies or diseconomies of scale. A positive linear combination of the Euclidean and rectangular distance measures, which we term the weighted one-two norm, is introduced. This distance function provides a linear regression model with interesting implications on the characterization of transportation networks. A directional bias function is defined, and examined in detail for the Pp and weighted one-two norms. In the second part of this study, several properties are derived for various forms of the continuous minisum location model. The Weiszfeld iterative solution procedure for the standard Weber problem with fp distances is also examined, and global and local convergence results obtained. These results are extended to the mixed-norm problem. In addition, optimality criteria are derived at non-differentiable points of the objective function.