On the feasible set for the squared Euclidean Weber problem and applications (original) (raw)

A projected Weiszfeld algorithm for the box-constrained Weber location problem

Applied Mathematics and Computation, 2011

The Weber problem consists of finding a point in R n that minimizes the weighted sum of distances from m points in R n that are not collinear. An application that motivated this problem is the optimal location of facilities in the 2-dimensional case. A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed-point iteration.

Single-facility Weber Location Problem based on the Lift Metric

Arxiv preprint arXiv:1105.0757, 2011

We investigate the continuous single-facility min-sum Weber location problem based on the lift and French metro metric. A problem with the lift metric or French metro metric (after transformation of Cartesian coordinates into polar coordinate system) is decomposed into a series of Weber problems based on the rectangular (Manhattan) metric, and at most two of such problems are solved with a standard procedure for the l 1 metric. An improvement algorithm based on the Weber problem with rectangular metric is proposed. Analogous improvements of the algorithms based on the lift and French metro metric are introduced. Two simple numerical examples are given. The asymptotic computational complexity of the algorithm proposed is estimated and proved by numerical experiments on large-scale problem examples. The comparison of the computational complexity of the proposed algorithm with the existing algorithm for equivalent problems formulated on graphs is given. AMS Subj. Class.: 90B85, 90B80

On the Weber facility location problem with limited distances and side constraints

Optimization Letters, 2012

The objective in the continuous facility location problem with limited distances is to minimize the sum of distance functions from the facility to the customers, but with a limit on each of the distances, after which the corresponding function becomes constant. The problem has applications in situations where the service provided by the facility is insensitive after a given threshold distance. In this paper, we propose a global optimization algorithm for the case in which there are in addition lower and upper bounds on the numbers of customers served.

Exact and approximate heuristics for the rectilinear Weber location problem with a line barrier

Computers & Operations Research, 2021

In this article, we propose an extension of the multi-Weber facility location problem with rectilinear-distance in the presence of passages over a non-horizontal line barrier. For the single-facility case, we develop an exact heuristic based on a divide-and-conquer approach that outperforms alternative heuristics available in literature. The multiple facilities case is solved by means of the application of an alternate-location-allocation heuristic heuristic, characterized by embedded exact and approximate procedures. For large instances, we propose a heuristic (with polynomial time complexity) which provides near-optimal solutions in a short computational time and a negligible gap. Finally, for testing purposes, we use a benchmark based on the transformation of the main problem into an equivalent p-median problem. Experimental results evidence the efficiency and validity of the proposed heuristics, which are capable of obtaining high quality solutions within acceptable computation times.

Algorithm for Constrained Weber Problem with Feasible Region Bounded by Arcs

The author proposes a heuristic algorithm for a special case of constrained continuous planar facility location problem with the connected region of feasible solutions bounded by arcs. The algorithm implements a modified Weiszfeld procedure and a special procedure for searching for the closest feasible solution for a given point. Example problems were solved. The convergence of the algorithm is proved experimentally for randomly generated problems. The results were compared with the results of the sample search.

Algebraic optimization: The Fermat-Weber location problem

Mathematical Programming, 1990

The Fermat-Weber location problem is to find a point in ~" that minimizes the sum of the (weighted) Euclidean distances from m given points in N". In this work we discuss some relevant complexity and algorithmic issues. First, using Tarski's theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that an e-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.

Algorithm for Weber Problem with a Metric Based on the Initial Fare

Journal of applied mathematics & informatics

We introduce a non-Euclidean metric for transportation systems with a defined minimum transportation cost (initial fare) and investigate the continuous single-facility Weber location problem based on this metric. The proposed algorithm uses the results for solving the Weber problem with Euclidean metric by Weiszfeld procedure as the initial point for a special local search procedure. The results of local search are then checked for optimality by calculating directional derivative of modified objective functions in finite number of directions. If the local search result is not optimal then algorithm solves constrained Weber problems with Euclidean metric to obtain the final result. An illustrative example is presented.

The profit-maximizing weber problem

Location Science, 1995

We consider the location of a profit-maximizing. single-plant firm in a continuous space with a finite number of local markets. Five different spatial price policies are considered: (i) spatial discriminatory pricing. (ii) uniform delivered pricing. (iii) uniform mill pricing. (iv) zone pricing. and (v) mixed pricing where clients have the choice between pickup or delivery. Global optimization methods are proposed for solving the corresponding problems and numerical experiments are reported on. Kr~~ords: Firm location. spatial price policies. global optimization.