A Generalized Weiszfeld method for the Multi-Facility Location Problem

a b s t r a c t An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities

Feedback algorithm for switch location : analysis of complexity and application to network design

An accelerated feedback algorithm to solve the single-facility minisum problem is studied with application to designing networks with the star topology. The algorithm, in which the acceleration with respect to the Weiszfeld procedure is achieved by multiplying the current Weiszfeld iterate by an accelerating feedback factor, is shown to converge faster than the accelerating procedures available in the literature. Singularities encountered in the algorithm are discussed in detail. A simple practical exception handling subroutine is developed. Several applications of the algorithm to designing computer networks with the star topology are demonstrated. Applications of the algorithm as a subroutine for multi-switch location problems are considered. Various engineering aspects involved in acquiring and processing coordinates for geographic locations are discussed. A complete algorithm in pseudocode along with the source code listing in Mathematica 4.1 is presented

A generalization of the rectangular bounding method for continuous location models

AbstractWe develop a generalized bounding method for the Weiszfeld iterative procedure used to solve the hyperbolically approximated ℓp-norm single- and multifacility minimum location problems. We also show that, at optimality, the solution to the bound problem coincides with the solution to the original location problem. We use this result to show that the rectangular bound value converges to the single-facility location problem optimal objective function value

Generalized Weiszfeld algorithms for Lq optimization

In many computer vision applications, a desired model of some type is computed by minimizing a cost function based on several measurements. Typically, one may compute the model that minimizes the L₂ cost, that is the sum of squares of measurement errors with respect to the model. However, the Lq solution which minimizes the sum of the qth power of errors usually gives more robust results in the presence of outliers for some values of q, for example, q = 1. The Weiszfeld algorithm is a classic algorithm for finding the geometric L1 mean of a set of points in Euclidean space. It is provably optimal and requires neither differentiation, nor line search. The Weiszfeld algorithm has also been generalized to find the L1 mean of a set of points on a Riemannian manifold of non-negative curvature. This paper shows that the Weiszfeld approach may be extended to a wide variety of problems to find an Lq mean for 1 ≤ q <; 2, while maintaining simplicity and provable convergence. We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global Lq optimum) and multiple rotation averaging (for which no such proof exists). Experimental results of Lq optimization for rotations show the improved reliability and robustness compared to L₂ optimization.This research has been funded by National ICT Australia

Iterative methodology on locating a cement plant

In this study, a cement plant location was determined by considering essential parameters such as the locations of resources and their importance in the manufacturing process. A crucial mathematical problem, named Weber problem, reinforced the decision of the method of allocating the factory. Additionally, not only the limitations of the cement production but also the importance weights of goods used in the manufacturing were taken into account in the iterative methodology in order to answer the engineering question via the mathematical problem. As a result, by optimizing the case through the iterations introduced in the paper, the location of the cement plant was set. Hence several losses such as extra travel distances and time wasting in transportation were minimized.No sponso