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

On the ramified optimal allocation problem

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.
Citation: Qinglan Xia, Shaofeng Xu. On the ramified optimal allocation problem. Networks and 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.

[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.

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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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

[24]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

[24]

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

[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.

[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.

[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.

[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.

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