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

Alessio Brancolini; Giuseppe Buttazzo; Filippo Santambrogio; Eugene Stepanov

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 3, page 509-524
- ISSN: 1292-8119

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topBrancolini, Alessio, et al. "Long-term planning versus short-term planning in the asymptotical location problem." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2008): 509-524. <http://eudml.org/doc/90924>.

@article{Brancolini2008,

abstract = {
Given the probability measure ν over the given region
$\Omega\subset \mathbb\{R\}^n$, we consider the optimal location of a set
Σ composed by n points in Ω in order to minimize the
average distance $\Sigma\mapsto \int_\Omega \mathrm\{dist\}\,(x,\Sigma)\,\{\rm d\}\nu$ (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
$n\to\infty$, although the optimization costs in both cases have the same asymptotic
orders of vanishing.
},

author = {Brancolini, Alessio, Buttazzo, Giuseppe, Santambrogio, Filippo, Stepanov, Eugene},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Location problem; facility location; Fermat-Weber problem; k-median problem; sequential allocation; average distance functional; optimal transportation; location problem; Fermat-weber problem; -median problem},

language = {eng},

month = {5},

number = {3},

pages = {509-524},

publisher = {EDP Sciences},

title = {Long-term planning versus short-term planning in the asymptotical location problem},

url = {http://eudml.org/doc/90924},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Brancolini, Alessio

AU - Buttazzo, Giuseppe

AU - Santambrogio, Filippo

AU - Stepanov, Eugene

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/5//

PB - EDP Sciences

VL - 15

IS - 3

SP - 509

EP - 524

AB -
Given the probability measure ν over the given region
$\Omega\subset \mathbb{R}^n$, we consider the optimal location of a set
Σ composed by n points in Ω in order to minimize the
average distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (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
$n\to\infty$, although the optimization costs in both cases have the same asymptotic
orders of vanishing.

LA - eng

KW - Location problem; facility location; Fermat-Weber problem; k-median problem; sequential allocation; average distance functional; optimal transportation; location problem; Fermat-weber problem; -median problem

UR - http://eudml.org/doc/90924

ER -

## References

top- L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.19 (1995) 191–246. Zbl0957.49029
- L. Ambrosio, N. Gigli and G. Savarè, Gradient flows in metric spaces and in the spaces of probability measures, Lectures in Mathematics. ETH Zurich, Birkhäuser (2005). Zbl1090.35002
- G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal. C. R. Acad. Sci. Paris Ser. I335 (2002) 1–6.
- A. Brancolini and G. Buttazzo, Optimal networks for mass transportation problems. ESAIM: COCV11 (2005) 88–101. Zbl1103.49002
- G. Buttazzo and E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem. Ann. Scuola Norm. Sup. Cl. Sci.II (2003) 631–678. Zbl1127.49031
- G. Buttazzo and E. Stepanov, Minimization problems for average distance functionals, in Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara Ed., Quaderni di Matematica14, Seconda Universita di Napoli (2004) 47–83. Zbl1089.49040
- G. Buttazzo, E. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, Progress in Nonlinear Differential Equations and their Applications51. Birkhäuser (2002) 41–65. Zbl1055.49029
- G. Dal Maso, An introduction to Γ-convergence. Birkhauser, Basel (1992).
- L. Fejes Töth, Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der Math. Wiss. 65. Springer-Verlag, Berlin (1953). Zbl0052.18401
- F. Morgan and R. Bolton, Hexagonal economic regions solve the location problem. Amer. Math. Monthly109 (2002) 165–172. Zbl1026.90059
- S.J.N. Mosconi and P. Tilli, Γ-convergence for the irrigation problem. J. Convex Anal.12 (2005) 145–158. Zbl1076.49024
- E. Paolini and E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in ${\mathbb{R}}^{n}$. J. Math. Sciences (N.Y.)122 (2004) 105–122. Zbl1099.49029
- F. Santambrogio and P. Tilli, Blow-up of optimal sets in the irrigation probem. J. Geom. Anal.15 (2005) 343–362. Zbl1115.49029
- E. Stepanov, Partial geometric regularity of some optimal connected transportation networks. J. Math. Sciences (N.Y.)132 (2006) 522–552.
- A. Suzuki and Z. Drezner, The p-center location. Location Sci.4 (1996) 69–82. Zbl0917.90233
- A. Suzuki and A. Okabe, Using Voronoi diagrams, in Facility location: a survey of applications and methods, Z. Drezner Ed., Springer Series in Operations Research, Springer Verlag (1995) 103–118.
- T. Suzuki, Y. Asami and A. Okabe, Sequential location-allocation of public facilities in one- and two-dimensional space: comparison of several policies. Math. Program. Ser. B52 (1991) 125–146. Zbl0734.90051
- . URIhttp://cvgmt.sns.it/papers/brabutsan06/

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