Abstract
This chapter reviews and extends some basic principles. An extension of an early theorem by Mills proves that the directional derivative is a saddle point of the differential Lagrangian over the primal-dual optimality region. This suggests a linear extrapolation for response estimation, at least for rim data, whose bias can be predictable due to the underlying convexity structure. The theory of compatibility for rim data perturbations is reviewed with a simpler proof of its most fundamental theorem, and this is extended to the sensitivity of average, rather than marginal, prices. Beyond the rim, some attention is given to matrix coefficient perturbation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bradley, S.P., A.C. Hax and T.L. Magnanti: Applied Mathematical Programming, Addison-Wesley, Reading, MA 1977
Daellenbach, H.G., E.J. Bell: User’s Guide to Linear Programming, Prentice-Hall, Englewood Cliffs, NJ 1970
Dantzig, G.B.: Linear Programming and Extensions, Princeton University Press, Princeton, NJ 1963
Gal, T.: Shadow prices and sensitivity analysis in linear programming under degeneracy: A state of the art survey, Operations Research Spektrum 8(1986)59–71
Gal, T.: Postoptimal Analyses, Parametric Programming and Related Topics: Degeneracy, Multicriteria Decision Making Redundancy, 2nd. ed., Walter de Gruyter, Berlin 1995
Gass, S.I.: Linear Programming, 5th ed., McGraw-Hill, New York, NY 1985
Gilford, R.: An analysis of degeneracy, Ph.D. Thesis, Mathematics Department, University of Colorado at Denver, Denver, CO 1994
Goldman, A.J., A.W. Tucker: “Theory of linear programming”. In:Kuhn, H.W., A.W. Tucker (eds.): Linear Inequalities and Related Systems, Annals of Mathematical Studies Number 38, Princeton University Press, Princeton, NJ 1956, 53–97
Greenberg, H.J.: An analysis of degeneracy, Naval Logistics Research Quarterly 33(1986)635–655
Greenberg, H.J.: “The ANALYZE rulebase”. In: Mathematical Models for Decision Support, G. Mitra, H.J. Greenberg, FA. Lootsma, M.J. Rijckaert, and H-J. Zimmermann (eds.), Proceedings of NATO ASI, July 26-August 6, Springer-Verlag, Berlin 1988, 229–238
Greenberg, H.J.: How to analyze results of linear programs, Part 2: price interpretation, Interfaces 23:5(1993)97–114
Greenberg, H.J.: A Computer-Assisted Analysis System for Mathematical Programming Models and Solutions: A User’s Guide for ANALYZE, Kluwer Academic Publishers, Norwell, MA 1993
Greenberg, H.J.: Enhancements of ANALYZE: A computer-assisted analysis system for mathematical programming models and solutions, ACM Transactions On Mathematical Software 19:2(1993)233–256
Greenberg, H.J.: The ANALYZE rulebase for supporting LP analysis, Annals of Operations Research (to appear 1996)
Greenberg, H.J., F.H. Murphy: Modeling the national energy plan, Journal of the Operational Research Society 31(1980)965–973
Greenberg, H.J., F.H. Murphy: Computing market equilibria with price regulations using mathematical programming, Operations Research 33(1985)935–954
Kuhn, H.W., A.W. Tucker (eds.): Linear Inequalities and Related Systems, Annals of Mathematical Studies Number 38, Princeton University Press, Princeton, NJ 1956
Mills, H.D.: “Marginal values of matrix games and linear programs”. In: Kuhn, H.W., A.W. Tucker (eds.): Linear Inequalities and Related Systems, Annals of Mathematical Studies Number 38, Princeton University Press, Princeton, NJ 1956, 183–193
Rockafellar, R.T.: Convex Analysis, Princeton University Press, Princeton, NJ 1970
Rubin, D.S., H.M. Wagner: Shadow prices: Tips and traps for managers, Interfaces 20:4(1990)150–157
Ward, J.E., R.E. Wendell: Approaches to sensitivity analysis in linear programming, Annals of Operations Research 27(1990)3–38
Wendell, R.E.: The tolerance approach to sensitivity analysis in linear programming, Management Science 31(1985)564–578
Williams, H.P.: Model Building in Mathematical Programming, 3rd ed., Wiley-Interscience, New York, NY 1990
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Greenberg, H.J. (1997). Linear Programming 1: Basic Principles. In: Gal, T., Greenberg, H.J. (eds) Advances in Sensitivity Analysis and Parametic Programming. International Series in Operations Research & Management Science, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6103-3_3
Download citation
DOI: https://doi.org/10.1007/978-1-4615-6103-3_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7796-2
Online ISBN: 978-1-4615-6103-3
eBook Packages: Springer Book Archive