The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Linear Programming

  • George B. Dantzig
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_849

Abstract

An article by George Dantzig, the ‘father of linear programming’. The problem of minimizing or maximizing a function of several variables subject to constraints when all the functions are linear is called a ‘linear program’. Linear programs can be used to approximate the broad class of convex functions commonly encountered in economic planning. Thousands of linear programs are efficiently solved with the simplex method, an algorithm. Solving a model with alternative activities requires software not only for solving on computers large systems of equations but also for selecting the best combination from an astronomical number of possible combinations of activities.

Keywords

Bimatrix games Convex program Dantzig, G. Decomposition principle Dantzig, G. B. Kantorovich, L. V. Koopmans, T. C. KuhnTucker conditions Lagrange multipliers Leontief input–output model Linear programming Mathematical programs Mini-max theorem Mixed strategies Simplex method for solving linear programs von Neumann, J. 
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Bibliography

  1. Beale, E.M.L. 1954. Linear programming by the method of leading variables. Report of the Conference on Linear Programming, May, arranged by Ferranti Ltd, London.Google Scholar
  2. Benders, J.F. 1962. Partitioning procedures for solving mixed-variable programming problems. Numerische Mathematik 4: 238–252.CrossRefGoogle Scholar
  3. Charnes, A., W.W. Cooper, and B. Mellon. 1952. Blending aviation gasolines – A study in programming interdependent activities in an integrated oil company. Econometrica 20 (2): 135–159.CrossRefGoogle Scholar
  4. Dantzig, G.B. 1948. Programming in a linear structure. Washington, DC: Comptroller, USAF.Google Scholar
  5. Dantzig, G. 1949, 1951. Programming of interdependent activities, II, mathematical model. In Activity analysis of production and allocation, ed. T.C. Koopmans, 330–335. New York: Wiley; also published in Econometrica 17(3 and 4), 1949, 200–211.Google Scholar
  6. Dantzig, G.B. 1963. Linear programming and extensions. Princeton: Princeton University Press.CrossRefGoogle Scholar
  7. Dantzig, G.B., and P. Wolfe. 1960. A decomposition principle for linear programs. Operations Research 8 (1): 101–111.CrossRefGoogle Scholar
  8. Kantorovich, L.V. 1939. Mathematical methods in the organization and planning of production. Publication House of the Leningrad State University. Translated in Management Science 6 (1960), 366–422.Google Scholar
  9. Koopmans, T.C., ed. 1951. Activity analysis of production and allocation. New York: Wiley.Google Scholar
  10. Kuhn, H.W. and Tucker, A.W. 1951. [The symposium was held in 1950, but the proceedings volume was published in 1951.] Nonlinear programming. In Proceedings of the second Berkeley symposium on mathematical statistics and probability, ed. J. Neyman. Berkeley: University of California Press, 481–492; also in Econometrica 19(1) (1951), 50–51 (abstract).Google Scholar
  11. Leontief, W. 1951. The structure of the American economy, 1919–1939. New York: Oxford University Press.Google Scholar
  12. von Neumann, J. 1928. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, 295–320. Translated by Sonya Bargmann in contributions to the theory of games, vol. 4, ed. A.W. Koopmans and R.D. Luce, Annals of Mathematics study no. 40. Princeton: Princeton University Press, 1959, pp. 13–42.Google Scholar
  13. von Neumann, J. 1937. Über ein ökonomisches Gleichungssytem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines Mathematischen Kolloquiums No. 8. Translated in Review of Economic Studies 13 (1), (194546) 1–9.Google Scholar
  14. von Neumann, J. 1947. Discussion of a maximization problem. Manuscript, Institute for Advanced Study. Princeton.Google Scholar
  15. von Neumann, J. 1963. In Collected works, ed. A.H. Taub, vol. VI, 89–95. Oxford: Pergamon Press.Google Scholar
  16. Vajda, S. 1956. The theory of games and linear programming. New York: Wiley.Google Scholar
  17. Wood, M.K. and Dantzig, G.B. 1949, 1951. The programming of interdependent activities: general discussion. In Activity analysis of production and allocation, ed. T.C. Koopmans, 15–81. New York: Wiley, 1951, also in Econometrica 17(3 and 4) (1949), 193–199.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • George B. Dantzig
    • 1
  1. 1.