Numerical solutions of the mass transfer problem

Serge Dubuc; Issa Kagabo

RAIRO - Operations Research (2006)

  • Volume: 40, Issue: 1, page 1-17
  • ISSN: 0399-0559

Abstract

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Let μ and ν be two probability measures on the real line and let c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ whose marginals coincide with μ and ν, and whose total cost ∫∫ c(x,y)dξ(x,y) is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical methods and determine the convergence rate.

How to cite

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Dubuc, Serge, and Kagabo, Issa. "Numerical solutions of the mass transfer problem." RAIRO - Operations Research 40.1 (2006): 1-17. <http://eudml.org/doc/249746>.

@article{Dubuc2006,
abstract = { Let μ and ν be two probability measures on the real line and let c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ whose marginals coincide with μ and ν, and whose total cost ∫∫ c(x,y)dξ(x,y) is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical methods and determine the convergence rate. },
author = {Dubuc, Serge, Kagabo, Issa},
journal = {RAIRO - Operations Research},
keywords = {Continuous programming; transportation; mass transfer; optimization.; continuous programming; transportation; optimization},
language = {eng},
month = {7},
number = {1},
pages = {1-17},
publisher = {EDP Sciences},
title = {Numerical solutions of the mass transfer problem},
url = {http://eudml.org/doc/249746},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Dubuc, Serge
AU - Kagabo, Issa
TI - Numerical solutions of the mass transfer problem
JO - RAIRO - Operations Research
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 1
SP - 1
EP - 17
AB - Let μ and ν be two probability measures on the real line and let c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ whose marginals coincide with μ and ν, and whose total cost ∫∫ c(x,y)dξ(x,y) is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical methods and determine the convergence rate.
LA - eng
KW - Continuous programming; transportation; mass transfer; optimization.; continuous programming; transportation; optimization
UR - http://eudml.org/doc/249746
ER -

References

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  11. L. Kantorovitch, On the translocation of masses. Doklady Akad. Nauk. SSSR37 (1942) 199–201.  
  12. H.G. Kellerer, Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete67 (1984) 399–432.  
  13. V.L. Levin and A.A. Milyutin, The Problem of Mass Transfer with a Discontinuous Cost Function and the Mass Statement of the Duality for Convex Extremal Problems. Uspehi Mat. Nauk.34 (1979) 3–68.  
  14. G. Monge, Mémoire sur la théorie des déblais et des remblais. Mém. Math. Phys. Acad. Royale Sci., Paris (1781) 666–704.  
  15. S.T. Rachev and L. Rüschendorf, Solution of some transportation problems with relaxed or additional constraints SIAM J. Control Optim.32 (1994), 673–689.  
  16. A.H. Tchen, Inequalities for distributions with given marginals Ann. Prob.8 (1980) 814–827.  

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