Two Stage Stochastic Linear Programs

  • Julia L. Higle
  • Suvrajeet Sen
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 8)

Abstract

Over the past several decades, linear programming (LP) has established itself as one of the most fundamental tools for planning. Its applications have become routine in several disciplines including those within engineering, business, economics, environmental studies and many others. One may attribute this wide spread acceptance to: (a) an understanding of the power and scope of LP among practitioners, (b) good algorithms, and (c) widely available and reliable software. Furthermore, research on specialized problems (e.g. assignment, transportation, networks etc.) has made LP methodology indispensible to numerous industries including transportation, energy, manufacturing and telecommunications, to name a few. Notwithstanding its success, we note that traditional LP models are deterministic models. That is, all objective function and constraint coefficients are assumed to be known with precision. The assumption that all model parameters are known with certainty serves to limit the usefulness of the approach when planning under uncertainty.

Keywords

Transportation Lost Aircrafts Mili 

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References

  1. Askin, R.G., R Ritchie, and A. Krisht [1989], Product and manufacturing system design, 15 th Conf on Prod. Res. and Tech., Univ. of California, Berkeley.Google Scholar
  2. Beale, E.M. [1955], On minimizing a convex function subject to linear inequalities, Journal of the Royal Statistical Society, 17B, pp. 173–184.MathSciNetGoogle Scholar
  3. Benders, J.F. [1962], Partitioning procedures for solving mixed variables programming problems, Numerische Mathematik, 4, pp. 238–252.MathSciNetCrossRefMATHGoogle Scholar
  4. Birge, J.R. [1982], The value of the stochastic solution in stochastic linear programs with fixed recourse, Mathematical Programming, 24, pp. 314–325.MathSciNetCrossRefMATHGoogle Scholar
  5. Birge, J.R. [1985], Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33, pp. 989–1007.MathSciNetCrossRefMATHGoogle Scholar
  6. Carino, D.R., T. Kent, D.H. Myers, C. Stacy, M. Sylvanus, A. Turner, K. Watanabe and W.T. Ziemba [1994], The Russell-Yasuda Kasai financial planning model, Interfaces, 24, pp. 29–49.CrossRefGoogle Scholar
  7. Dantzig, G.B. [1955], Linear programming under uncertainty, Management Science, 1, pp. 197–206.MathSciNetCrossRefMATHGoogle Scholar
  8. Eppen G.D., R.K. Martin, and L. Schräge [1989], A scenario approach to capacity planning, Operations Research, 37, pp. 517–527.CrossRefGoogle Scholar
  9. Ferguson, A.R. and G.B. Dantzig [1957], The allocation of aircraft to routes, Management Science, vol. 3, pp. 45–73.MathSciNetCrossRefGoogle Scholar
  10. Frantzeskakis, L. and W.B. Powell [1990], A successive linear approximation procedure for stochastic dynamic vehicle allocation problems, Transportation Science, 24, pp. 40–57.MathSciNetCrossRefMATHGoogle Scholar
  11. Higle, J.L. and S. Sen [1991], Stochastic Decomposition: an algorithm for two-stage linear programs with recourse, Mathematics of Operations Research, 16, pp. 650–669.MathSciNetCrossRefMATHGoogle Scholar
  12. Kali, P. and S.W. Wallace [1994], Stochastic Programming, John Wiley and Sons, Chichester, England.Google Scholar
  13. Louveaux F.V. and Y. Smeers [1988], Optimal investment for electricity generation: A stochastic model and a test problem, in: Numerical Techniques for Stochastic Optimization, Y. Ermoliev and R. J-B. Wets (eds.), (Springer-Verlag, Berlin) pp. 445–453.CrossRefGoogle Scholar
  14. Masse, P. and R. Gibrat [1957], Application of linear programming to investments in the electric power industry, Management Science, 3, 149–166.CrossRefGoogle Scholar
  15. Mulvey, J.M. and A. Ruszczynski [1992], A new scenario decomposition method for large scale stochastic optimization, to appear in Operations Research.Google Scholar
  16. Mulvey, J.M. and H. Vladimirou [1989], Stochastic network optimization for investment planning, Annals of Operations Research, 20, pp. 187–217.MathSciNetCrossRefMATHGoogle Scholar
  17. Murphy, F.H., S. Sen, and A.L. Soyster [1982], Electric utility capacity expansion with uncertain load forecasts, IIE Transactions 14, pp. 52–59.CrossRefGoogle Scholar
  18. Pereira, M.V.F. and L.M.V.G. Pinto [1991], Multi-stage stochastic optimization applied to energy planning, Mathematical Programming, 52, pp. 359–375.MathSciNetCrossRefMATHGoogle Scholar
  19. Pinter, J. [1991], Stochastic modeling and optimization for environmental management, Annals of Operations Research, 31, pp. 527–544.MathSciNetCrossRefMATHGoogle Scholar
  20. Sen, S., R.D. Doverspike and S. Cosares [1994], Network planning with random demand, Telecommunications Systems, 3, pp. 11–30.CrossRefGoogle Scholar
  21. Somlyödy, L. and R. J-B. Wets [1988], Stochastic optimization models for lake eutrophication management, Operations Research, 36, pp. 660–681.MathSciNetCrossRefGoogle Scholar
  22. Walkup, D. and R. J-B. Wets [1969], Lifting projections of convex polyhedra, Pacific Journal of Mathematics, 28, pp. 465–475.MathSciNetCrossRefMATHGoogle Scholar
  23. Wets, R.J-B. [1974], Stochastic programs with fixed recourse: the equivalent deterministic program, SIAM Review 16 pp. 309–339.MathSciNetCrossRefMATHGoogle Scholar
  24. Wets, R. J-B [1983], Solving stochastic programs with simple recourse, Stochastics, 10, pp. 219–242.MathSciNetCrossRefMATHGoogle Scholar
  25. Wets, R. J-B. [1989], Stochastic Programming, in: Handbooks in Operations Research: Optimization, G.L. Nemhauser, A.H.G. Rinnoy Kan and M.J. Todd (eds.), North-Holland, pp. 573–629Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Julia L. Higle
    • 1
  • Suvrajeet Sen
    • 1
  1. 1.University of ArizonaUSA

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