Skip to main content Accessibility help
×
Home

Mass transportation problems with capacity constraints

  • S. T. Rachev (a1) and I. Olkin (a2)

Abstract

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. L p -distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

Copyright

Corresponding author

Postal address: Statistics and Applied Probability, University of California, Santa Barbara, CA 93106–3110, USA.
∗∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305–4065, USA. Supported in part by National Science Foundation.

Footnotes

Hide All

Supported in part by Econometric Institute, Erasmus University, Rotterdam and the Alexander von Humboldt Foundation.

Footnotes

References

Hide All
Appell, P. (1884). Mémoires sur les deblais et remblais des systèmes continués et discontinués. Mémoires des sav. etr., Ser. 2, 29, 3 mémoire.
Balinski, M. L., and Rachev, S. T. (1989). On Monge–Kantorovich problems. Preprint, SUNY, Stony Brook, Dept. of Applied Mathematics & Statistics. NSF-Grant Proposal, DMS-89 02330.
Barnes, E. R., and Hoffman, A. J. (1985). On transportation problems with upper bounds on leading rectangles. SIAM J. Alg. Discrete Methods 6, 487496.
Cambanis, S., Simons, G., and Stout, W. (1976). Inequalities for Ek(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitsth. 36, 285294.
Cuesta-Albertos, J. A., Matran, C., Rachev, S. T. and Rüschendorf, L. (1996). Mass transportation problems in probability theory. Math. Scientist 21, 3772.
Dall'Aglio, G. (1956). Sugli estremi dei momenti delle funzioni di ripartizione dopia. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, 3374.
Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39, 15631572.
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks-Cole, Pacific Grove, CA.
Gini, C. (1914). Di una misura della dissomiglianza tra due gruppi di quantita e della sue applicazioni allo studio delle relazioni statistiche. Atti del Reale Instituo Veneto di Sci., Lettera et Arti. LXXIV (1914–1915), 185213.
Hoffman, A. J. (1963). On simple linear programming problems. In Convexity ed. Klee, V. (Proc. Symp. Pure Math. 7). AMS, Providence, RI, pp. 317327.
Hoffman, A. J., and Veinott, A. F. (1990). Staircase transportation problems with superadditive rewards and cumulative capacities. Preprint, Stanford University, Stanford, CA.
Hoffman, A. J., and Veinott, A. F. (1993). Staircase transportation problems with superadditive rewards and cumulative capacities. Math. Programming 62, 199213.
Kakosyan, A. B., Klebanov, L., and Rachev, S. T. (1988). Quantitative Criteria for Convergence of Probability Measures. Ayastan, Erevan. (In Russian.)
Kalashnikov, V. V., and Rachev, S. T. (1990). Mathematical Methods for Construction of Stochastic Queueing Models. Wadsworth & Brooks-Cole, Pacific Grove, CA.
Kantorovich, L. V. (1942). On the transfer of masses. Dokl. Acad. Nauk. USSR 37, 227229.
Kantorovich, L. V. (1948). On a problem of Monge. Usp. Mat. Nauk 3, 225226. (In Russian.)
Kantorovich, L. V., and Akilov, G. P. (1984). Functional Analysis, Nauka, Moscow. (In Russian.)
Kruskal, W. H. (1958). Ordinal measures of association. J. Amer. Statist. Assoc. 53, 814861.
Levin, V. L., and Rachev, S. T. (1989). New duality theorems for marginal problems with some applications in stochastics. In Lecture Notes in Math. 1412. Springer, New York, pp. 137170.
Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais. In Histoire de l'Academie Royale des Sciences avec les memoires de mathematique et physique pour la mème année, pp. 666704.
Olkin, I., and Rachev, S. T. (1990). Distributions with given marginals. Tech. Report 270, Dept. of Statistics, Stanford University, Stanford, CA.
Rachev, S. T. (1981). On minimal metrics in the space of real-valued random variables. Sov. Math. Dokl. 23, 425432.
Rachev, S. T. (1984). Hausdorff metric construction in the probability measures space. Pliska, Studia Mathematica 7, 152162.
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester.
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770799.
Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems; Vol. 1, Theory; Vol. 2, Applications. Springer, New York (to appear).
Stoyan, P. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.
Sudakov, V. N. (1976). Geometric problems in the theory of infinite-dimensional probability distributions. Tr. Mat. Inst. V. A. Steklov Akad. Nauk. SSSR 141. (In Russian.) (English translation (1979). Proc. Steklov Inst. Math. 2.)
Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8, 814827.
Topkis, D. M., and Veinott, A. F. Jr. (1973). Monotone solution of extremal problems on lattices (Abstract). In Abstracts of 8th International Symposium on Mathematical Programming. Stanford University, Stanford, CA, p. 131.
Veinott, A. F. Jr. (1989). Representation of general and polyhedral sublattices and sublattices of product spaces. Linear Alg. Appl. 114/115, 681704.

Keywords

MSC classification

Mass transportation problems with capacity constraints

  • S. T. Rachev (a1) and I. Olkin (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed