Abstract
We define the complete problem of a two-stage linear programming under uncertainty, to be:
where x is the first-stage decision variable, the pair (y+, y−) represents the second-stage decision variables. In order to solve this class of problem, we derive a convex programming problem, whose set of optimal solutions is identical to the set of optimal solutions of our original problem. This problem is called the equivalent convex programming. If the random variable ξ has a continuous distribution, we give an algorithm to solve the equivalent convex program. Moreover, we derive explicitly the equivalent convex program for a few common distributions.
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References
Beale, E. M.: The use of quadratic programming in stochastic linear programming. RAND P-2404, August 1961.
Dantzig, G. B.: Linear programming and extensions. Princeton: Princeton Univ. Press 1963.
El-Agizy, M.: Programming under uncertainty with discrete distribution function. Univ. of California, Operations Res. Center, ORC 64-13, 1964.
Elmaghraby, S. E.: An approach to linear programming under uncertainty. Operations Res. 7, No. 1, 208–216 (1959).
Miller, C. E.: The simplex method for local separable programming. In Recent advances in mathematical programming, pp. 89–100 (Ed. Graves and Wolfe) 1963.
Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1963.
Zoutendijk, G.: Methods of feasible directions. Amsterdam: Elsevier 1960.
Köhler, D., and R. Wets: Programming under uncertainty: An experimental code for the complete problem. Univ. of California, Berkeley ORC 64-15, 1964.
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Parts of this material was written while the author was at the Operations Research Center at the University of California (Berkeley), where this research was partially supported by the Office of Naval Research under Contract Number 222(83) with the University of California and by a research grant from the National Science Foundation.
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Wets, R. Programming under uncertainty: The complete problem. Z. Wahrscheinlichkeitstheorie verw Gebiete 4, 316–339 (1966). https://doi.org/10.1007/BF00539117
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DOI: https://doi.org/10.1007/BF00539117