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June  2013, 8(2): 591-624. doi: 10.3934/nhm.2013.8.591

On the ramified optimal allocation problem

Qinglan Xia 1, and  Shaofeng Xu 2,

1. 

Department of Mathematics, University of California at Davis, Davis, CA 95616, United States
2. 

Department of Economics, University of California at Davis, Davis, CA 95616, United States

Received  August 2011 Revised  October 2012 Published  May 2013

This paper proposes an optimal allocation problem with ramified transport technologies in a spatial economy. Ramified transportation is used to model network-like branching structures attributed to the economies of scale in group transportation. A social planner aims at finding an optimal allocation plan and an associated optimal allocation path to minimize the overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transport literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It is shown that due to the transport economies of scale, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study the properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.
Keywords: transport economies of scale, optimal transport path, branching structure, Ramified transportation, state matrix., assignment map, allocation problem.
Mathematics Subject Classification: Primary: 91B32, 58E17; Secondary: 49Q20, 90B1.
Citation: Qinglan Xia, Shaofeng Xu. On the ramified optimal allocation problem. Networks & Heterogeneous Media, 2013, 8 (2) : 591-624. doi: 10.3934/nhm.2013.8.591
References:
[1]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matematiques, 49 (2005), 417-451. doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[2]

M. Bernot, V. Caselles and J.-M. Morel, "Optimal Transportation Networks. Models and Theory," Lecture Notes in Mathematics, Vol. 1955, Springer-Verlag, Berlin, 2009.  Google Scholar

[3]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces, Journal of the European Mathematical Society, 8 (2006), 415-434. doi: 10.4171/JEMS/61.  Google Scholar

[4]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function, Interfaces and Free Boundaries, 13 (2011), 191-222. doi: 10.4171/IFB/254.  Google Scholar

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G. Buttazzo and G. Carlier, Optimal spatial pricing strategies with transportation costs, in "Nonlinear Analysis and Optimization II. Optimization," Contemporary Mathematics, 514, Amer. Math. Soc., Providence, RI, (2010), 105-121. doi: 10.1090/conm/514/10102.  Google Scholar

[6]

G. Carlier and I. Ekeland, Matching for teams, Economic Theory, 42 (2010), 397-418. doi: 10.1007/s00199-008-0415-z.  Google Scholar

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P.-A. Chiappori, R. McCann and L. Nesheim, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness, Economic Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.  Google Scholar

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G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1-49.  Google Scholar

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I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types, Economic Theory, 42 (2010), 275-315. doi: 10.1007/s00199-008-0427-8.  Google Scholar

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A. Figalli, Y.-H. Kim and R. McCann, When is multidimensional screening a convex program?, Journal of Economic Theory, 146 (2011), 454-478. doi: 10.1016/j.jet.2010.11.006.  Google Scholar

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E. N. Gilbert, Minimum cost communication networks, Bell System Technical Journal, 46 (1967), 2209-2227. Google Scholar

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L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.), 37 (1942), 199-201.  Google Scholar

[13]

F. Maddalena, S. Solimini and J.-M. Morel, A variational model of irrigation patterns, Interfaces and Free Boundaries, 5 (2003), 391-415. doi: 10.4171/IFB/85.  Google Scholar

[14]

R. McCann and M. Trokhimtchouk, Optimal partition of a large labor force into working pairs, Economic Theory, 42 (2010), 375-395. doi: 10.1007/s00199-008-0420-2.  Google Scholar

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G. Monge, Mémoire sur la théorie des déblais et de remblais, Histoire de l'Académie Royale des Sciences de Paris, avec les Mémorires de Mathématique et de Physique pour la même année, (1781), 666-704. Google Scholar

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

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F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. doi: 10.4171/IFB/160.  Google Scholar

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C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.  Google Scholar

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C. Villani, "Optimal Transport. Old and New," Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[20]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Mathematics, 5 (2003), 251-279. doi: 10.1142/S021919970300094X.  Google Scholar

[21]

Q. Xia, Interior regularity of optimal transport paths, Calculus of Variations and Partial Differential Equations, 20 (2004), 283-299. doi: 10.1007/s00526-003-0237-6.  Google Scholar

[22]

Q. Xia, The formation of tree leaf, ESAIM Control, Optimisation and Calculus of Variations, 13 (2007), 359-377. doi: 10.1051/cocv:2007016.  Google Scholar

[23]

Q. Xia, The geodesic problem in quasimetric spaces, Journal of Geometric Analysis, 19 (2009), 452-479. doi: 10.1007/s12220-008-9065-4.  Google Scholar

[24]

Q. Xia, Boundary regularity of optimal transport paths, Advances in Calculus of Variations, 4 (2011), 153-174. doi: 10.1515/ACV.2010.024.  Google Scholar

[25]

Q. Xia, Ramified optimal transportation in geodesic metric spaces, Advances in Calculus of Variations, 4 (2011), 277-307. doi: 10.1515/ACV.2011.002.  Google Scholar

[26]

Q. Xia, Numerical simulation of optimal transport paths, in "The Second International Conference on Computer Modeling and Simulation," 1, IEEE, (2010), 521-525. doi: 10.1109/ICCMS.2010.30.  Google Scholar

[27]

Q. Xia and A. Vershynina, On the transport dimension of measures, SIAM Journal on Mathematical Analysis, 41 (2010), 2407-2430. doi: 10.1137/090770205.  Google Scholar

[28]

Q. Xia and S. Xu, The exchange value embedded in a transport system, Applied Mathematics and Optimization, 62 (2010), 229-252. doi: 10.1007/s00245-010-9102-0.  Google Scholar

show all references

References:
[1]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matematiques, 49 (2005), 417-451. doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[2]

M. Bernot, V. Caselles and J.-M. Morel, "Optimal Transportation Networks. Models and Theory," Lecture Notes in Mathematics, Vol. 1955, Springer-Verlag, Berlin, 2009.  Google Scholar

[3]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces, Journal of the European Mathematical Society, 8 (2006), 415-434. doi: 10.4171/JEMS/61.  Google Scholar

[4]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function, Interfaces and Free Boundaries, 13 (2011), 191-222. doi: 10.4171/IFB/254.  Google Scholar

[5]

G. Buttazzo and G. Carlier, Optimal spatial pricing strategies with transportation costs, in "Nonlinear Analysis and Optimization II. Optimization," Contemporary Mathematics, 514, Amer. Math. Soc., Providence, RI, (2010), 105-121. doi: 10.1090/conm/514/10102.  Google Scholar

[6]

G. Carlier and I. Ekeland, Matching for teams, Economic Theory, 42 (2010), 397-418. doi: 10.1007/s00199-008-0415-z.  Google Scholar

[7]

P.-A. Chiappori, R. McCann and L. Nesheim, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness, Economic Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.  Google Scholar

[8]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1-49.  Google Scholar

[9]

I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types, Economic Theory, 42 (2010), 275-315. doi: 10.1007/s00199-008-0427-8.  Google Scholar

[10]

A. Figalli, Y.-H. Kim and R. McCann, When is multidimensional screening a convex program?, Journal of Economic Theory, 146 (2011), 454-478. doi: 10.1016/j.jet.2010.11.006.  Google Scholar

[11]

E. N. Gilbert, Minimum cost communication networks, Bell System Technical Journal, 46 (1967), 2209-2227. Google Scholar

[12]

L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.), 37 (1942), 199-201.  Google Scholar

[13]

F. Maddalena, S. Solimini and J.-M. Morel, A variational model of irrigation patterns, Interfaces and Free Boundaries, 5 (2003), 391-415. doi: 10.4171/IFB/85.  Google Scholar

[14]

R. McCann and M. Trokhimtchouk, Optimal partition of a large labor force into working pairs, Economic Theory, 42 (2010), 375-395. doi: 10.1007/s00199-008-0420-2.  Google Scholar

[15]

G. Monge, Mémoire sur la théorie des déblais et de remblais, Histoire de l'Académie Royale des Sciences de Paris, avec les Mémorires de Mathématique et de Physique pour la même année, (1781), 666-704. Google Scholar

[16]

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

[17]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. doi: 10.4171/IFB/160.  Google Scholar

[18]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[19]

C. Villani, "Optimal Transport. Old and New," Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[20]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Mathematics, 5 (2003), 251-279. doi: 10.1142/S021919970300094X.  Google Scholar

[21]

Q. Xia, Interior regularity of optimal transport paths, Calculus of Variations and Partial Differential Equations, 20 (2004), 283-299. doi: 10.1007/s00526-003-0237-6.  Google Scholar

[22]

Q. Xia, The formation of tree leaf, ESAIM Control, Optimisation and Calculus of Variations, 13 (2007), 359-377. doi: 10.1051/cocv:2007016.  Google Scholar

[23]

Q. Xia, The geodesic problem in quasimetric spaces, Journal of Geometric Analysis, 19 (2009), 452-479. doi: 10.1007/s12220-008-9065-4.  Google Scholar

[24]

Q. Xia, Boundary regularity of optimal transport paths, Advances in Calculus of Variations, 4 (2011), 153-174. doi: 10.1515/ACV.2010.024.  Google Scholar

[25]

Q. Xia, Ramified optimal transportation in geodesic metric spaces, Advances in Calculus of Variations, 4 (2011), 277-307. doi: 10.1515/ACV.2011.002.  Google Scholar

[26]

Q. Xia, Numerical simulation of optimal transport paths, in "The Second International Conference on Computer Modeling and Simulation," 1, IEEE, (2010), 521-525. doi: 10.1109/ICCMS.2010.30.  Google Scholar

[27]

Q. Xia and A. Vershynina, On the transport dimension of measures, SIAM Journal on Mathematical Analysis, 41 (2010), 2407-2430. doi: 10.1137/090770205.  Google Scholar

[28]

Q. Xia and S. Xu, The exchange value embedded in a transport system, Applied Mathematics and Optimization, 62 (2010), 229-252. doi: 10.1007/s00245-010-9102-0.  Google Scholar

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