Asymptotic optimal location of facilities in a competition between population and industries

Buttazzo, Giuseppe; Santambrogio, Filippo; Stepanov, Eugene

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 239-273.

We consider the problem of optimally locating a given number $k$ of points in ${ℝ}^n$ for an integral cost function which takes into account two measures ${\varphi }^{+}$ and ${\varphi }^{-}$. The points represent for example new industrial facilities that have to be located, the measure ${\varphi }^{+}$ representing in this case already existing industries that want to be close to the new ones, and ${\varphi }^{-}$ representing private citizens who want to stay far away. The asymptotic analysis as $k\to \infty$ is performed, providing the asymptotic density of optimal locations.

Classification : 49Q20, 49Q10

[1] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. | MR | Zbl

[2] L. Ambrosio and C. Mantegazza, Curvature and distance function from a manifold, J. Geom. Anal. 8 (1998), 723–748. Dedicated to the memory of Fred Almgren. | MR | Zbl

[3] L. Ambrosio and P. Tilli, “Topics on Analysis in Metric Spaces”, Vol. 25, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004. | MR | Zbl

[4] G. Bouchitté, C. Jimenez and R. Mahadevan, Asymptotic analysis of a class of optimal location problems, J. Math. Pures Appl. 95 (2011), 382–419. | MR | Zbl

[5] G. Bouchitté, C. Jimenez and R. Mahadevan, Asymptotique d’un problème de positionnement optimal, C. R. Math. Acad. Sci. Paris 335 (2002), 853–858. | MR | Zbl

[6] A. Brancolini, G. Buttazzo, F. Santambrogio and E. Stepanov, Long-term planning versus short-term planning in the asymptotical location problem, ESAIM Control Optim. Calc. Var. 15 (2009), 509–524. | EuDML | Numdam | MR | Zbl

[7] G. Buttazzo, Optimization problems in mass transportation theory, Boll. Unione Mat. Ital. 1 (2008), 401–427. | MR | Zbl

[8] P. Cohort, Limit theorems for random normalized distortion, Ann. Appl. Probab. 14 (2004), 118–143. | MR | Zbl

[9] G. Dal Maso, “An Introduction to $\Gamma$-convergence”, Birkhäuser, Basel, 1992. | MR | Zbl

[10] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842–850. | MR | Zbl

[11] S. P. Fekete, J. S. B. Mitchell and K. Beurer, On the continuous Fermat-Weber problem, Oper. Res. 53 (2005), 61–76. | MR | Zbl

[12] S. Graf and H. Luschgy, “Foundations of Quantization for Probability Distributions”, Vol. 1730, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. | MR | Zbl

[13] D. Liberzon, “Switching in Systems and Control”, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 2003. | MR | Zbl

[14] C. Mantegazza and A. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim. 47 (2003), 1–25. | MR | Zbl

[15] F. Morgan and R. Bolton, Hexagonal economic regions solve the location problem, Amer. Math. Monthly 109 (2002), 165–172. | MR | Zbl

[16] S. J. N. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Convex Anal. 12 (2005), 145–158. | MR | Zbl

[17] A. Suzuki and Z. Drezner, The $p$-center location, Location Science 4 (1996), 69–82. | Zbl

[18] 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, 103–118, Springer Verlag, 1995. | MR

[19] A. Weber, “ Über den Standort der Industrien, Erster Teil: Reine Theorie des Standortes”, Mohr, Tübingen, 1909.