Optimal location problems with routing cost. (English) Zbl 1275.49076
Summary: In the paper a model problem for the location of a given number \(N\) of points in a given region \(\Omega\) and with a given resources density \(\rho(x)\) is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra routing cost is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as \(N \to \infty\) is characterized and some applications to airfreight systems are shown.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 90B80 | Discrete location and assignment |
| 90B85 | Continuous location |
Keywords:
location problems; routing cost; \(\Gamma\)-convergence; transport problems; airfreight systemsReferences:
| [1] | K. A. Al Kaabi, “The Geography of Airfreight and Metropolitan Economies: Potential Connection,”, Ph.D. thesis (2010) |
| [2] | G. Bouchitté, <em>Asymptotique d’un problème de positionnement optimal</em>,, C. R. Acad. Sci. Paris, 335, 853 (2002) · Zbl 1041.90025 · doi:10.1016/S1631-073X(02)02575-X |
| [3] | A. Brancolini, <em>Long-term planning versus short-term planning in the asymptotical location problem</em>,, ESAIM Control Optim. Calc. Var., 15, 509 (2009) · Zbl 1169.90386 · doi:10.1051/cocv:2008034 |
| [4] | G. Buttazzo, <em>Optimal transportation problems with free Dirichlet regions</em>,, In, 51, 41 (2002) · Zbl 1055.49029 |
| [5] | G. Buttazzo, <em>A mass transportation model for the optimal planning of an urban region</em>,, SIAM Rev., 51, 593 (2009) · Zbl 1170.49013 · doi:10.1137/090759197 |
| [6] | G. Buttazzo, <em>Asymptotic optimal location of facilities in a competition between population and industries</em>,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12, 239 (2013) · Zbl 1264.49047 |
| [7] | G. Buttazzo, <em>Asymptotics of an optimal compliance-location problem</em>,, ESAIM Control Optim. Calc. Var., 12, 752 (2006) · Zbl 1114.49016 · doi:10.1051/cocv:2006020 |
| [8] | P. Cohort, <em>Limit theorems for random normalized distortion</em>,, Ann. Appl. Prob., 14, 118 (2004) · Zbl 1041.60022 · doi:10.1214/aoap/1075828049 |
| [9] | G. Dal Maso, “An Introduction to \(\Gamma \)-Convergence,”, Progress in Nonlinear Differential Equations and their Applications (PNLDE), 8 (1993) · Zbl 0816.49001 · doi:10.1007/978-1-4612-0327-8 |
| [10] | L. Fejes Tóth, “Lagerungen in der Ebene, auf der Kugel und im Raum,”, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete (1953) · Zbl 0052.18401 |
| [11] | A. Hofton, <em>The identification of the airfreight operating cost parameters for the use in the Sika-Samgods freight model</em>,, Tech. report (2002) |
| [12] | J. P. Johnson, <em>Air cargo operations cost database</em>,, NASA/CR-1998-207655, 1998 (1985) |
| [13] | F. Morgan, <em>Hexagonal economic regions solve the location problem</em>,, Amer. Math. Monthly, 109, 165 (2002) · Zbl 1026.90059 · doi:10.2307/2695328 |
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