Optimizing Facility Location with Euclidean and Rectilinear Distances

Introduction

In continuous location theory, facilities to be optimally located are generally represented by points, and the customers or markets that they serve are also geometrical points in space. The objective is to find the optimal site of one or more facilities with respect to a specified performance measure such as the sum of transportation costs. This is one of the oldest formal optimization problems in mathematics and has a long and interesting history ([15] Section 1.3, [7,9,21]). Many variants of the problem exist. A very basic version of the location problem is to minimize:

$$ W(x) = \sum\limits_{i=1}^n {w_i K(x-a_i )} \:, $$

(1)

where $x = (x_{1}, x_{2})$ is the unknown facility location in $\Re^2$, w _i  is a positive weight representing transportation cost per unit distance for customer i, and $K(x-a_{i})$ is a norm measuring distance from the facility location x to the fixed location $a_{i} =(a_{i1},a_{i2})$ of demand point i. The most common distance...