An iterative curve fitting approach for solving the Weber problem in spatial economics

Abstract

Early methods for solving the Weber problem by locating the point of minimum aggregate distance employed physical analogues. There is no closed-form mathematical method for replicating these mechanical procedures because analytical procedures result in high order polynomials requiring numerical methods. Iterative techniques using gradient related methods can be used; but in the small number of cases where a trial solution coincides with a data point or where the final solution itself is a data point, gradient methods are unable to reach a solution. Other common iterative methods, which are not gradient related, avoid these difficulties, but are less efficient.

The method presented in algorithm form does not encounter difficulty when a trial solution encounters a data point. A paraboloid is fitted through five points on the surface formed by the total distances and derivatives are used to locate a trial minimum. The trial minimum becomes the center of the next paraboloid and the process is continued. The algorithm presented here is simpler to program and run than the gradient related methods, when they are combined with a separate test for the conditions of a minimum. In addition, the algorithm is more efficient than the non-gradient related methods such as the grid search technique.