Abstract
In this paper we present a method for the solution of a one stage stochastic programming problem, where the underlying problem is an LP and some of the right hand side values are random variables. The stochastic programming problem that we formulate contains probabilistic constraint and penalty, incorporated into the objective function, used to penalize violation of the stochastic constraints. We solve this problem by a dual type algorithm. The special case where only penalty is used while the probabilistic constraint is disregarded, the simple recourse problem, was solved earlier by Wets, using a primal simplex algorithm with individual upper bounds. Our method appears to be simpler. The method has applications to nonstochastic programming problems too, e.g., it solves the constrained minimum absolute deviation problem.
Zusammenfassung
In dieser Arbeit wird eine Methode vorgestellt zur Lösung einstufiger stochastischer Programme, wobei das zugrundeliegende Problem ein LP mit zufälligen rechten Seiten darstellt. Das resultierende stochastische Programm enthält Wahrscheinlichkeitsrestriktionen und Strafterme, letztere innerhalb der Zielfunktion zur Bestrafung von Abweichungen in den stochastischen Restriktionen. Wir lösen dieses Problem mit einem dualen Algorithmus. Der Spezialfall, in dem ausschließlich Strafterme benutzt werden und Wahrscheinlichkeitsrestriktionen unberücksichtigt bleiben, d.h. das einfache Kompensationsmodell, wurde bereits früher von Wets mittels eines primalen Simplex-Algorithmus mit einzelnen oberen Schranken gelöst. Unsere Methode scheint einfacher zu sein. Die Methode ist auch auf nicht-stochastische Programme anwendbar, z.B. auf das Problem minimaler absoluter Abweichungen von Nebenbedingungen.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E.M.L. Beale (1955) On Minimizing a Convex Function Subject to Linear Inequalities, Journal of the Royal Statistical Society Ser. B. 17, 173–184
R.G. Bland (1977) New Finite Pivoting Rules for the Simplex Method, Mathematics of Operations Research 2, 103–107
A. Charnes, W.W. Cooper and G.H. Symonds (1985) Cost Horizons and Certainty Equivalents: An Approach to Stochastic Programming of Heating Oil Production, Management Science 4, 236–263
G.B. Dantzig (1955) Linear Programming under Uncertainty, Management Science 1, 197–206
C.E. Lemke (1954) The Dual Method for Solving the Linear Programming Problem, Naval Research Logistic Quarterly 1, 36–47.
K. Murty (1968) Linear Programming under Uncertainty: A Basic Property of the Optimal Solution, Zeitschrift für Wahrscheinlichkeitstheorie ver. Geb. 10, 284–288
A. Prékopa (1973) Contributions to the Theory of Stochastic Programming, Mathematical Programming 4, 202–221
A. Prékopa (1990) The Discrete Moment Problem and Linear Programming. To appear in Discrete Applied Mathematics
R.J.B. Wets (1983) Solving Stochastic Programs with Simple Recourse, Stochastics 10, 219–242
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Prékopa, A. Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution. ZOR - Methods and Models of Operations Research 34, 441–461 (1990). https://doi.org/10.1007/BF01421551
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01421551