Skip to main content
SpringerLink
Log in

The Simplex Method

  • Chapter
Linear and Nonlinear Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 228))

  • 278k Accesses

Abstract

The idea of the simplex method is to proceed from one basic feasible solution (that is, one extreme point) of the constraint set of a problem in standard form to another, in such a way as to continually decrease the value of the objective function until a minimum is reached. The results of Chap. 2 assure us that it is sufficient to consider only basic feasible solutions in our search for an optimal feasible solution. This chapter demonstrates that an efficient method for moving among basic solutions to the minimum can be constructed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. R.H. Bartels, A numerical investigation of the simplex method. Technical Report No. CS 104, Computer Science Department, Stanford University, Stanford, CA (31 July 1968)

    Google Scholar 

  2. R.H. Bartels, G.H Golub, The simplex method of linear programming using LU decomposition. Commun. ACM 12(5), 266–268 (1969)

    Google Scholar 

  3. M.S. Bazaraa, J.J. Jarvis, Linear Programming and Network Flows (Wiley, New York, 1977)

    Google Scholar 

  4. G. Birkhoff, Three observations on linear algebra. Rev. Univ. Nac. Tucumán, Ser. A. 5, 147–151 (1946)

    Google Scholar 

  5. R.E. Bixby, Progress in linear programming. ORSA J. Comput. 6(1), 15–22 (1994)

    Article  Google Scholar 

  6. R.G. Bland, New finite pivoting rules for the simplex method. Math. Oper. Res. 2(2), 103–107 (1977)

    Article  Google Scholar 

  7. A. Charnes, Optimality and degeneracy in linear programming. Econometrica 20, 160–170 (1952)

    Article  Google Scholar 

  8. A. Charnes, C.E. Lemke, The bounded variables problem. ONR Research Memorandum 10, Graduate School of Industrial Administration, Carnegie Institute of Technology, Pittsburgh (1954)

    Google Scholar 

  9. G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities (Chap. XXI), in Activity Analysis of Production and Allocation, ed. by T.C. Koopmans. Cowles Commission Monograph, vol. 13 (Wiley, New York, 1951)

    Google Scholar 

  10. G.B., Dantzig, Application of the simplex method to a transportation problem, in Activity Analysis of Production and Allocation, ed. by T.C. Koopmans (Wiley, New York, 1951), pp. 359–373

    Google Scholar 

  11. G.B. Dantzig, Computational algorithm of the revised simplex method. RAND Report RM-1266, The RAND Corporation, Santa Monica, CA (1953)

    Google Scholar 

  12. G.B. Dantzig, Variables with upper bounds in linear programming. RAND Report RM-1271, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  13. G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, 1963)

    Google Scholar 

  14. G.B. Dantzig, A. Orden, P. Wolfe, Generalized simplex method for minimizing a linear form under linear inequality restraints. RAND Report RM-1264, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  15. G.B. Dantzig, P. Wolfe, Decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)

    Article  Google Scholar 

  16. G.E. Forsythe, C.B. Moler, Computer Solution of Linear Algebraic Systems (Prentice-Hall, Englewood Cliffs, NJ, 1967)

    Google Scholar 

  17. D. Gale, The Theory of Linear Economic Models (McGraw-Hill, New York, 1960)

    Google Scholar 

  18. F.L. Hitchcock, The distribution of a product from several sources to numerous localities. J. Math. Phys. 20, 224–230 (1941)

    Google Scholar 

  19. T.C. Koopmans, Optimum utilization of the transportation system, in Proceedings of the International Statistical Conference, Washington, DC, 1947

    Google Scholar 

  20. B.A. Murtagh, Advanced Linear Programming (McGraw-Hill, New York, 1981)

    Google Scholar 

  21. K.G. Murty, Linear and Combinatorial Programming (Wiley, New York, 1976)

    Google Scholar 

  22. W. Orchard-Hays, Background development and extensions of the revised simplex method. RAND Report RM-1433, The RAND Corporation, Santa Monica, CA (1954)

    Google Scholar 

  23. A. Orden, Application of the simplex method to a variety of matrix problems, in Directorate of Management Analysis: “Symposium on Linear Inequalities and Programming”, ed. by A. Orden, L. Goldstein (DCS/Comptroller, Headquarters, U.S. Air Force, Washington, DC, 1952), pp. 28–50

    Google Scholar 

  24. M. Simonnard, Linear Programming, translated by William S. Jewell (Prentice-Hall, Englewood Cliffs, NJ, 1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management Science and Engineering, Stanford University, Stanford, CA, USA

    David G. Luenberger & Yinyu Ye

Authors
  1. David G. Luenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinyu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Luenberger, D.G., Ye, Y. (2016). The Simplex Method. In: Linear and Nonlinear Programming. International Series in Operations Research & Management Science, vol 228. Springer, Cham. https://doi.org/10.1007/978-3-319-18842-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation