Long-term planning versus short-term planning in the asymptotical location problem

¹ SISSA, 4 via Beirut, 34014 Trieste, Italy; brancoli@sissa.it
² Dipartimento di Matematica, Università di Pisa, 5 Largo B. Pontecorvo, 56127 Pisa, Italy; buttazzo@dm.unipi.it
³ CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France; santambrogio@ceremade.dauphine.fr
⁴ Dipartimento di Matematica, Università di Pisa, 5 Largo B. Pontecorvo, 56127 Pisa, Italy; stepanov.eugene@gmail.com

Received: 28 February 2007

Abstract

Given the probability measure ν over the given region , we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as , although the optimization costs in both cases have the same asymptotic orders of vanishing.