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April  2014, 34(4): 1301-1317. doi: 10.3934/dcds.2014.34.1301

Optimal location problems with routing cost

Giuseppe Buttazzo 1, , Serena Guarino Lo Bianco 2, and  Fabrizio Oliviero 3,

1. 

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa
2. 

Dipartimento di Matematica - Università di Pisa, Largo B. Pontecorvo, 5, 56127 Pisa, Italy
3. 

Dipartimento di Ingegneria Aerospaziale - Università di Pisa, Via G. Caruso, 8, 56122 Pisa, Italy

Received  August 2012 Revised  November 2012 Published  October 2013

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.
Keywords: routing cost, airfreight systems., Gamma-convergence, Location problems, transport problems.
Mathematics Subject Classification: Primary: 49Q10, 49Q20; Secondary: 90B80, 90B8.
Citation: Giuseppe Buttazzo, Serena Guarino Lo Bianco, Fabrizio Oliviero. Optimal location problems with routing cost. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1301-1317. doi: 10.3934/dcds.2014.34.1301
References:
[1]

K. A. Al Kaabi, "The Geography of Airfreight and Metropolitan Economies: Potential Connection," Ph.D. thesis, University of North Carolina, 2010. Google Scholar

[2]

G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris, 335 (2002), 853-858. doi: 10.1016/S1631-073X(02)02575-X.  Google Scholar

[3]

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. doi: 10.1051/cocv:2008034.  Google Scholar

[4]

G. Buttazzo, E. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, In "Variational Methods for Discontinuous Structures,'' Progress in Nonlinear Differential Equations Appl., 51, Birkhäuser Verlag, Basel, (2002), 41-65.  Google Scholar

[5]

G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region, SIAM Rev., 51 (2009), 593-610. doi: 10.1137/090759197.  Google Scholar

[6]

G. Buttazzo, F. Santambrogio and E. Stepanov, Asymptotic optimal location of facilities in a competition between population and industries, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 239-273. Google Scholar

[7]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769. doi: 10.1051/cocv:2006020.  Google Scholar

[8]

P. Cohort, Limit theorems for random normalized distortion, Ann. Appl. Prob., 14 (2004), 118-143. doi: 10.1214/aoap/1075828049.  Google Scholar

[9]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications (PNLDE), 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[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, Band LXV, Springer-Verlag, Berli-Göttingen-Heidelbergn, 1953.  Google Scholar

[11]

A. Hofton, The identification of the airfreight operating cost parameters for the use in the Sika-Samgods freight model, Tech. report, 2002. Google Scholar

[12]

J. P. Johnson and E. M. Gaier, Air cargo operations cost database, NASA/CR-1998-207655, Tech. report, 1985. Google Scholar

[13]

F. Morgan and R. Bolton, Hexagonal economic regions solve the location problem, Amer. Math. Monthly, 109 (2002), 165-172. doi: 10.2307/2695328.  Google Scholar

show all references

References:
[1]

K. A. Al Kaabi, "The Geography of Airfreight and Metropolitan Economies: Potential Connection," Ph.D. thesis, University of North Carolina, 2010. Google Scholar

[2]

G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris, 335 (2002), 853-858. doi: 10.1016/S1631-073X(02)02575-X.  Google Scholar

[3]

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. doi: 10.1051/cocv:2008034.  Google Scholar

[4]

G. Buttazzo, E. Oudet and E. Stepanov, Optimal transportation problems with free Dirichlet regions, In "Variational Methods for Discontinuous Structures,'' Progress in Nonlinear Differential Equations Appl., 51, Birkhäuser Verlag, Basel, (2002), 41-65.  Google Scholar

[5]

G. Buttazzo and F. Santambrogio, A mass transportation model for the optimal planning of an urban region, SIAM Rev., 51 (2009), 593-610. doi: 10.1137/090759197.  Google Scholar

[6]

G. Buttazzo, F. Santambrogio and E. Stepanov, Asymptotic optimal location of facilities in a competition between population and industries, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 239-273. Google Scholar

[7]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769. doi: 10.1051/cocv:2006020.  Google Scholar

[8]

P. Cohort, Limit theorems for random normalized distortion, Ann. Appl. Prob., 14 (2004), 118-143. doi: 10.1214/aoap/1075828049.  Google Scholar

[9]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications (PNLDE), 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[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, Band LXV, Springer-Verlag, Berli-Göttingen-Heidelbergn, 1953.  Google Scholar

[11]

A. Hofton, The identification of the airfreight operating cost parameters for the use in the Sika-Samgods freight model, Tech. report, 2002. Google Scholar

[12]

J. P. Johnson and E. M. Gaier, Air cargo operations cost database, NASA/CR-1998-207655, Tech. report, 1985. Google Scholar

[13]

F. Morgan and R. Bolton, Hexagonal economic regions solve the location problem, Amer. Math. Monthly, 109 (2002), 165-172. doi: 10.2307/2695328.  Google Scholar

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