Linear Programming

Part of the Springer Optimization and Its Applications book series (SOIA, volume 36)


The simplest and most widely spread models of convex programming are linear programming models; in other words, models with linear objective function and with linear constraints. This might turn out to be a serious restriction on our field of interest. But as shown in Chapter 1, a wide variety of problems can be satisfactorily represented by linear models. In many cases, the problem naturally takes a linear form; in some cases where this is not so, the problem may be approximately represented by a linear model. As mentioned by Vandermeulen [37, p. 4], “At least in the initial stages, linear models yield more economic output from less mathematical input.” In the preface to their well-known book, Dorfman, Samuelson, and Solow [12] denote linear programming as “one of the most important postwar developments in economic theory” [12, p. vii].


Extreme Point Dual Problem Linear Programming Problem Simplex Method Shadow Price 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Further Reading

  1. [1]
    J. Arrous, The Leontief pollution model: A systematic formulation, Econ. Syst. Res., 6 (1994), 105–107.CrossRefGoogle Scholar
  2. [2]
    K. J. Arrow, George Dantzig in the development of economic analysis, Discrete Optim., 5 (2008), 159–167.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    M. L. Balinski and W. J. Baumol, The dual in nonlinear programming and its economic interpretation, Rev. Econ. Stud., 35-3 (1968), 237–256.MATHCrossRefGoogle Scholar
  4. [4]
    W. J. Baumol and R. C. Bushnell, Error produced by linearization in mathematical programming, Econometrica, 35-3–4 (1967), 447–471.CrossRefGoogle Scholar
  5. [5]
    M. J. Beckmann, Comparative statics in linear programming and the Giffen paradox, Rev. Econ. Stud., 23 (1956), 232–235.Google Scholar
  6. [6]
    A. Charnes, S. Duffuaa, and M. Ryan, The more for less paradox in linear programming, Eur. J. Oper. Res., 31 (1987), 194–197.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Charnes and D. Klingman, The more-for-less paradox in the distribution model, Cahiers Centre Études Rech. Opér., 13-1 (1971), 11–22.MATHMathSciNetGoogle Scholar
  8. [8]
    H. B. Chenery and W. J. Raduchel, Substitution in planning models, in H. B. Chenery, ed., Studies in Development Planning, Harvard University Press, Cambridge, MA, 1971.Google Scholar
  9. [9]
    M. Chobot and F. Turnovec, O jednom ekonomickom paradoxe v modeloch lineárneho programovania (One economic paradox in linear programming models), Ekon.-Mat. Obzor, 4 (1974), 376–386.MathSciNetGoogle Scholar
  10. [10]
    V. Chvátal, Linear Programming,W. H. Freeman, New York, 1983.MATHGoogle Scholar
  11. [11]
    G. B. Dantzig and W. Orchard-Hays, The product form for the inverse in the simplex method, Math. Tables Other Aids Comput., 8 (1954), 64–67.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    R. Dorfman, P. A. Samuelson, and R. M. Solow, Linear Programming and Economic Analysis, McGraw–Hill, New York, 1958.MATHGoogle Scholar
  13. [13]
    E. Fermi, Thermodynamics, Dover, New York, 1937.Google Scholar
  14. [14]
    G. Gandolfo, International Trade Theory and Policy, Springer-Verlag, Berlin, 1998.Google Scholar
  15. [15]
    D. Gale, The Theory of Linear Economic Models, McGraw–Hill, New York, 1960.Google Scholar
  16. [16]
    S. I. Gass, Linear Programming: Methods and Applications, 3rd ed., McGraw–Hill, New York, 1969.Google Scholar
  17. [17]
    S. I. Gass, Linear Programming: Methods and Applications, 4th ed., McGraw–Hill, New York, 1975.MATHGoogle Scholar
  18. [18]
    N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373–395.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    L. G. Khachian, A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR, 224 (1979), 1093–1096 (in Russian); Sov. Math. Dokl., 20 (1979), 191–194 (English translation).Google Scholar
  20. [20]
    V. Klee and G.J. Minty, How good is the simplex algorithm?, in O. Shisha, ed., Inequalities, Vol. III, Academic Press, New York, 1972, 159–175.Google Scholar
  21. [21]
    G. Leblanc and P. V. Moeseke, The Le Chatelier principle in convex programming, Rev. Econ. Stud., 43-133 (1976), 143–147.MATHGoogle Scholar
  22. [22]
    W. W. Leontief, Environmental repercussions and the economic structure: An input– output approach, Rev. Econ. Statist., 52 (1970), 262–271.CrossRefGoogle Scholar
  23. [23]
    P. D. Lowe, Pricing problems in an input–output approach to environmental protection, Rev. Econ. Statist., 61 (1979), 110–117.CrossRefGoogle Scholar
  24. [24]
    M. Luptáčik, A note on the “more-for-less” paradox, in P. Kischka, U. Leopold- Wildburger, R. H. Möhring, and F. J. Rademacher, eds., Models, Methods and Decision Support for Management: Essays in Honor of Paul Stähly, Physica-Verlag, Würtzburg, Germany, 2001.Google Scholar
  25. [25]
    M. Luptáčik and B. Böhm, Reconsideration of non-negative solutions for the augmented Leontief model, Econ. Syst. Res., 6 (1994), 167–170.CrossRefGoogle Scholar
  26. [26]
    M. Luptáčik and B. Böhm, A consistent formulation of the Leontief pollution model, Econ. Syst. Res., 11 (1999), 263–275.CrossRefGoogle Scholar
  27. [27]
    Y. Murata, Mathematics for Stability and Optimization of Economic Systems, Academic Press, New York, 1977.Google Scholar
  28. [28]
    D. Ricardo, Principles of Political Economy and Taxation, John Murray, London, 1817; reprint of the 3rd ed., Bell and Sons, London, 1924.Google Scholar
  29. [29]
    C. Ross, T. Terlaky, and J. Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach,Wiley, New York, 1997.Google Scholar
  30. [30]
    P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, MA, 1947.MATHGoogle Scholar
  31. [31]
    P. A. Samuelson, The Le Chatelier Principle in Linear Programming, Research memorandum, Rand Corporation, Santa Monica, CA, 1949.Google Scholar
  32. [32]
    E. Silberberg and W. Suen, The Structure of Economics: A Mathematical Analysis, 3rd ed., McGraw–Hill, Singapore, 2001.Google Scholar
  33. [33]
    A. E. Steenge, Environmental repercussions and the economic structure: Further comments, Rev. Econ. Statist., 60 (1978), 482–486.CrossRefGoogle Scholar
  34. [34]
    A. Takayama, International Trade: An Approach to the Theory, Holt, Reinhart, and Winston, New York, 1972.Google Scholar
  35. [35]
    Th. ten Raa, Linear Analysis of Competitive Economics, LSE Handbooks in Economics, Harvester Wheatsheaf, London, 1995.Google Scholar
  36. [36]
    Th. ten Raa, The Economics of Input–Output Analysis, Cambridge University Press, New York, 2005.Google Scholar
  37. [37]
    D. C. Vandermeulen, Linear Economic Theory, Prentice–Hall, Englewood Cliffs, NJ, 1971.Google Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Department of EconomicsVienna University of Economics and Business AdministrationViennaAustria

Personalised recommendations