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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

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