We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time; a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a linear-time constant-competitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, linear-time constant-factor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time.

Lagrangean relaxation is largely used to solve combinatorial optimization problems. A known problem for Lagrangean relaxation application is the definition of convenient step size control in subgradient like methods. Even preserving theoretical convergence properties, a wrong defined control can reflect in performance and increase computational times, a critical point in large scale instances. We show in this work how to accelerate a classical subgradient method, using the local information of the surrogate constraints relaxed in the Lagrangean relaxation. It results in a one-dimensional search that corrects the step size and is independent of the step size control used. The application to Capacitated and Uncapacitated Facility Location problems is shown. Several computational tests confirm the superiority of this scheme. Key words: Location problems, Lagrangean relaxation, Subgradient method. 1. Introduction Facility location is the problem of locating a number of facilities from a s...

In this paper we consider a location and pricing model for a retail #rm that wants to enter a spatial market where a competitor #rm is already operating as a monopoly with several outlets. The entering #rms seeks to determine the optimal uniform mill price and its servers' locations that maximizes pro#ts given the reaction in price of the competitor #rm to its entrance. A tabu seach procedure is presented to solve the model together with computational experience and an example. jel = C61, R32,L81,R12 keywords = competitive location, discrete network location, spatial pricing # Corresponding author: Daniel Serra, Department of Economics and Business, Universitat Pompeu Fabra, Trias Fargas 25-27, Barcelona 08005, Spain. Tel: 34-93-5421666, Fax: 34-93-5421746, e-mail: daniel.serra@econ.upf.es. Paper presented at the 44th North American Meetings of the Regional Science Association International, Bu#alo. y Dept. of Geography and Environmental Engineering, The Johns Hopkins University ...

In facility layout problems, a major concern is the optimal design or remodeling of the facilities of an organization. The decision maker’s objective is to arrange the facility in an optimal way, so that the interaction among functions (i.e. machines, inventories, persons) and places (i.e. offices, work locations, depots) is efficient. A simple pure-binary LP model is developed and solved for a small hospital, where five functions are to be redesigned to five different locations. The model is rather flexible and can be used, with small modifications, for larger facility layouts.