Encyclopedia of Operations Research and Management Science

2013 Edition
| Editors: Saul I. Gass, Michael C. Fu

Parametric Programming

  • Tomas Gal
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-1153-7_733


The meaning of a parameter as used here is best explained by a simple example. Recall that a parabola can be expressed as follows: y = ax2, a ≠ 0. Setting a = 1, a parabola is obtained that has a different shape from the parabola when setting, for example, a = 5. In both cases, however, there are parabolas that obey specific relationships; only the shapes are different. Hence, the parabola y = ax2 describes a family of parabolas and the parameter a specifies the shape.

Consider the general mathematical-programming problem:
$$ {\rm Max}\ z = f({x}) $$
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  1. Ashram, H. (2007). Construction of the largest sensitivity region for general linear programs. Applied Mathematics and Computation, 189, 1435–1447.CrossRefGoogle Scholar
  2. Bank, B., Guddat, J., Klatte, D., Kummer, B., & Tammer, T. (1982). Nonlinear parametric optimization. Berlin: Akademie Verlag.CrossRefGoogle Scholar
  3. Bradley, S. P., Hax, A. C., & Magnanti, T. L. (1977). Applied mathematical programming. Reading, MA: Addison-Wesley.Google Scholar
  4. Dawande, M. W., & Hooker, J. N. (2000). Inference-based sensitivity analysis for mixed integer/linear programming. Operations Research, 48, 623–634.CrossRefGoogle Scholar
  5. Dinkelbach, W. (1969). Sensitivitätsanalysen und parametrische Programmierung. Berlin: Springer Verlag.CrossRefGoogle Scholar
  6. Drud, A. S., & Lasdon, L. (1997). Nonlinear programming. In T. Gal & H. J. Greenberg (Eds.), Advances in sensitivity analysis and parametric programming. Norwell, MA: Kluwer.Google Scholar
  7. Faisca, N. P., Kosmidis, V. D., Rustem, B., & Pistikopoulos, E. N. (2009). Global optimization of multi-parametric MILP problems. Journal Global Optimization, 45(1), 131–151.CrossRefGoogle Scholar
  8. Filippi, C. (2005). A fresh view on the tolerance approach to sensitivity analysis in linear programming. European Journal of Operational Research, 167, 1–19.CrossRefGoogle Scholar
  9. Gal, T. (1973). Betriebliche Entscheidungsprobleme, Sensitivitätsanalyse und parametrische Programmierung. Berlin: W. de Gruyter.Google Scholar
  10. Gal, T. (1979). Postoptimal analyses, parametric programming and related topics. New York: McGraw Hill.Google Scholar
  11. Gal, T. (1980). A ‘historiogramme’ of parametric programming. Journal of the Operational Research Society, 31, 449–451.CrossRefGoogle Scholar
  12. Gal, T. (1983). A note on the history of parametric programming. Journal of the Operational Research Society, 34, 162–163.CrossRefGoogle Scholar
  13. Gal, T. (1993). Putting the LP survey into perspective. OR/MS Today, 19(6), 93.Google Scholar
  14. Gal, T. (1994a). Selected bibliography on degeneracy. Annals Operations Research.Google Scholar
  15. Gal, T. (1994b). Postoptimal analyses and parametric programming. Berlin: W. de Gruyter. Revised and updated edition.Google Scholar
  16. Gal, T., & Greenberg, H. J. (Eds.). (1997). Advances in sensitivity analysis and parametric programming. Norwell, MA: Kluwer.Google Scholar
  17. Greenberg, H. J. (1993). A computer-assisted analysis system for mathematical programming models and solutions: A user's guide for ANALYZE. Norwell, MA: Kluwer.CrossRefGoogle Scholar
  18. Gass, S. I. (1985). Linear programming (5th ed.). New York: McGraw-Hill.Google Scholar
  19. Gass, S. I., & Saaty, T. L. (1955). The parametric objective function. Naval Research Logistics Quarterly, 2, 39–45.CrossRefGoogle Scholar
  20. Guddat, J., Guerra Vazquez, F., & Jongen, H. T. (1991). Parametric optimization: Singularities, path following and jumps. Stuttgart/New York: B. G. Teubner/Wiley.Google Scholar
  21. Hadigheh, A. G., Mirnia, K., & Terlaky, T. (2007). Active constraint set invariancy sensitivity analysis in linear optimization. JOTA, 133, 303–315.CrossRefGoogle Scholar
  22. Hladik, M. (2008a). Additive and multiplicative tolerance in multiobjective linear programming. Operations Research Letters, 36, 393–396.CrossRefGoogle Scholar
  23. Hladik, M. (2008b). Computing the tolerance in multiobjective linear programming. Optimization Methods and Software, 23, 731–739.CrossRefGoogle Scholar
  24. Hladik, M. (2010). Multiparametric linear programming: Support set and optimal partition invariancy. European Journal of Operational Research, 202, 25–31.CrossRefGoogle Scholar
  25. Kheirfam, B. (2010). Sensitivity analysis in multi-parametric strictly convex quadratic optimization. Matem. Vesnik, 62, 95–107.Google Scholar
  26. Kruse, H.-J. (1986). Degeneracy graphs and the neighborhood problem (Lecture Notes in economics and mathematical systems No. 260). Berlin: Springer Verlag.CrossRefGoogle Scholar
  27. Manne, A. S. (1953). Notes on parametric linear programming, RAND Report P-468. Santa Monica, CA: The Rand Corporation.Google Scholar
  28. Ravi, N., & Wendell, R. E. (1988). Tolerance approach to sensitivity analysis in network linear programming. Networks, 18, 159–181.CrossRefGoogle Scholar
  29. Saaty, T. L., & Gass, S. I. (1954). The parametric objective function, Part I. Operations Research, 2, 316–319.Google Scholar
  30. Steuer, R. E. (1986). Multiple criteria optimization: Theory, computation, and application. New York: Wiley.Google Scholar
  31. Wendell, R. E. (1985). The tolerance approach to sensitivity analysis in linear programming. Management Science, 31, 564–578.CrossRefGoogle Scholar
  32. Wendell, R. E. (2004). Tolerance sensitivity and optimality bounds in linear programming. Management Science, 50, 797–803.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Fern Universität in HagenHagenGermany