Transportation Problem in Operations Research and its Applications
Keywords:
transportation, operations, research, applications, mathematical, economics
Abstract
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.[1]
In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".[2][3]
Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich.[4] Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem.[5] The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.[6]
References
1. G. Monge. Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666–704, 1781.
2. Schrijver, Alexander, Combinatorial Optimization, Berlin ; New York : Springer, 2003. ISBN 3540443894. Cf. p. 362
3. Ivor Grattan-Guinness, Ivor, Companion encyclopedia of the history and philosophy of the mathematical sciences, Volume 1, JHU Press, 2003. Cf. p.831
4. L. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199–201, 1942.
5. Cédric Villani (2003). Topics in Optimal Transportation. American Mathematical Soc. p. 66. ISBN 978-0-8218-3312-4.
6. Singiresu S. Rao (2009). Engineering Optimization: Theory and Practice (4th ed.). John Wiley & Sons. p. 221. ISBN 978-0-470-18352-6.
7. Frank L. Hitchcock (1941) "The distribution of a product from several sources to numerous localities", MIT Journal of Mathematics and Physics 20:224–230 MR0004469.
8. D. R. Fulkerson (1956) Hitchcock Transportation Problem, RAND corporation.
9. L. R. Ford Jr. & D. R. Fulkerson (1962) § 3.1 in Flows in Networks, page 95, Princeton University Press
10. L. Ambrosio, N. Gigli & G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)
11. Angenent, S.; Haker, S.; Tannenbaum, A. (2003). "Minimizing flows for the Monge–Kantorovich problem". SIAM J. Math. Anal. 35 (1): 61–97. CiteSeerX 10.1.1.424.1064. doi:10.1137/S0036141002410927.
12. Galichon, Alfred. Optimal Transport Methods in Economics. Princeton University Press, 2016.
13. Rachev, Svetlozar T., and Ludger Rüschendorf. Mass Transportation Problems: Volume I: Theory. Vol. 1. Springer, 1998.
14. Santambrogio, Filippo. Optimal Transport for Applied Mathematicians. Birkhäuser Basel, 2016. In particular chapter 6, section 4.2.
15. Aurenhammer, Franz (1987), "Power diagrams: properties, algorithms and applications", SIAM Journal on Computing, 16 (1): 78–96, doi:10.1137/0216006, MR 0873251.
16. Peyré, Gabriel and Marco Cuturi (2019), "Computational Optimal Transport: With Applications to Data Science", Foundations and Trends in Machine Learning: Vol. 11: No. 5-6, pp 355–607. DOI: 10.1561/2200000073.
17. Haker, Steven; Zhu, Lei; Tannenbaum, Allen; Angenent, Sigurd (1 December 2004). "Optimal Mass Transport for Registration and Warping". International Journal of Computer Vision. 60 (3): 225–240. CiteSeerX 10.1.1.59.4082. doi:10.1023/B:VISI.0000036836.66311.97. ISSN 0920-5691. S2CID 13261370.
18. Glimm, T.; Oliker, V. (1 September 2003). "Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem". Journal of Mathematical Sciences. 117 (3): 4096–4108. doi:10.1023/A:1024856201493. ISSN 1072-3374. S2CID 8301248.
19. Kasim, Muhammad Firmansyah; Ceurvorst, Luke; Ratan, Naren; Sadler, James; Chen, Nicholas; Sävert, Alexander; Trines, Raoul; Bingham, Robert; Burrows, Philip N. (16 February 2017). "Quantitative shadowgraphy and proton radiography for large intensity modulations". Physical Review E. 95 (2): 023306. arXiv:1607.04179. Bibcode:2017PhRvE.95b3306K. doi:10.1103/PhysRevE.95.023306. PMID 28297858. S2CID 13326345.
20. Metivier, Ludovic (24 February 2016). "Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion". Geophysical Journal International. 205 (1): 345–377. Bibcode:2016GeoJI.205.345M. doi:10.1093/gji/ggw014.
2. Schrijver, Alexander, Combinatorial Optimization, Berlin ; New York : Springer, 2003. ISBN 3540443894. Cf. p. 362
3. Ivor Grattan-Guinness, Ivor, Companion encyclopedia of the history and philosophy of the mathematical sciences, Volume 1, JHU Press, 2003. Cf. p.831
4. L. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199–201, 1942.
5. Cédric Villani (2003). Topics in Optimal Transportation. American Mathematical Soc. p. 66. ISBN 978-0-8218-3312-4.
6. Singiresu S. Rao (2009). Engineering Optimization: Theory and Practice (4th ed.). John Wiley & Sons. p. 221. ISBN 978-0-470-18352-6.
7. Frank L. Hitchcock (1941) "The distribution of a product from several sources to numerous localities", MIT Journal of Mathematics and Physics 20:224–230 MR0004469.
8. D. R. Fulkerson (1956) Hitchcock Transportation Problem, RAND corporation.
9. L. R. Ford Jr. & D. R. Fulkerson (1962) § 3.1 in Flows in Networks, page 95, Princeton University Press
10. L. Ambrosio, N. Gigli & G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)
11. Angenent, S.; Haker, S.; Tannenbaum, A. (2003). "Minimizing flows for the Monge–Kantorovich problem". SIAM J. Math. Anal. 35 (1): 61–97. CiteSeerX 10.1.1.424.1064. doi:10.1137/S0036141002410927.
12. Galichon, Alfred. Optimal Transport Methods in Economics. Princeton University Press, 2016.
13. Rachev, Svetlozar T., and Ludger Rüschendorf. Mass Transportation Problems: Volume I: Theory. Vol. 1. Springer, 1998.
14. Santambrogio, Filippo. Optimal Transport for Applied Mathematicians. Birkhäuser Basel, 2016. In particular chapter 6, section 4.2.
15. Aurenhammer, Franz (1987), "Power diagrams: properties, algorithms and applications", SIAM Journal on Computing, 16 (1): 78–96, doi:10.1137/0216006, MR 0873251.
16. Peyré, Gabriel and Marco Cuturi (2019), "Computational Optimal Transport: With Applications to Data Science", Foundations and Trends in Machine Learning: Vol. 11: No. 5-6, pp 355–607. DOI: 10.1561/2200000073.
17. Haker, Steven; Zhu, Lei; Tannenbaum, Allen; Angenent, Sigurd (1 December 2004). "Optimal Mass Transport for Registration and Warping". International Journal of Computer Vision. 60 (3): 225–240. CiteSeerX 10.1.1.59.4082. doi:10.1023/B:VISI.0000036836.66311.97. ISSN 0920-5691. S2CID 13261370.
18. Glimm, T.; Oliker, V. (1 September 2003). "Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem". Journal of Mathematical Sciences. 117 (3): 4096–4108. doi:10.1023/A:1024856201493. ISSN 1072-3374. S2CID 8301248.
19. Kasim, Muhammad Firmansyah; Ceurvorst, Luke; Ratan, Naren; Sadler, James; Chen, Nicholas; Sävert, Alexander; Trines, Raoul; Bingham, Robert; Burrows, Philip N. (16 February 2017). "Quantitative shadowgraphy and proton radiography for large intensity modulations". Physical Review E. 95 (2): 023306. arXiv:1607.04179. Bibcode:2017PhRvE.95b3306K. doi:10.1103/PhysRevE.95.023306. PMID 28297858. S2CID 13326345.
20. Metivier, Ludovic (24 February 2016). "Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion". Geophysical Journal International. 205 (1): 345–377. Bibcode:2016GeoJI.205.345M. doi:10.1093/gji/ggw014.
Published
2022-05-31
How to Cite
Meena, K. P. (2022). Transportation Problem in Operations Research and its Applications. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 3(5), 90-101. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/178
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