Abstract
Mathematical programming models have been applied to many problems in various fields. The data of real problems contain uncertainty and are thus represented as random variables. Stochastic programming deals with optimization under uncertainty. In Chap. 6, the basic model of the stochastic programming problem and solution algorithms are shown. Section 6.1 shows the basics of stochastic programming with recourse. This model is the most fundamental in the stochastic programming method, and its application to real problems is widely applied. In this model, when constraints are violated in accordance with probability fluctuations, new decisions are introduced to correct inconvenience. In this model, the total cost including the cost of additional decisions is minimised. The additional cost and its expected value are referred to as a recourse function, and it is shown that the recourse function becomes a convex function of the initial decision variable. In Sect. 6.2 we deal with the extension of stochastic programming with recourse to multistage planning. In multistage planning, problems generally include subproblems that correspond to a large number of scenarios. A basic method for decomposing such large scale problem is shown. Section 6.3 considers probabilistic programming including integer conditions. Especially the exact solution method for problem solving for probability planning problem including 0–1 variables. The 0–1 variable can be applied to actual problems such as various decision making, execution and interruption of specific decisions, and its application range is wide. Section 6.4 shows a stochastic programming model including stochastic constraints. This chance constraint gives the lower limit value of the probability of satisfying the constraint, and it has a deep relationship with the value at risk. We also consider the numerical integration method to calculate such probability. In Sect. 6.5, we consider stochastic programming considering risk. In particular, the problem considering variance of costs is dealt with. Since the objective function is a non-convex function, an exact solution by a branch cut method is presented which approximates objective function by a convex quadratic function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahmed, S., Tawarmalani, M., & Sahinidis, N. V. (2005). A finite branch-and-bound algorithm for two-stage stochastic integer programs. Mathematical Programming, 100, 355–377.
Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252.
Birge, J. R. (1985). Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33, 989–1007.
Birge, J. R. (1997). Stochastic programming computation and applications. INFORMS Journal on Computing, 9, 111–133.
Birge, J. R., Donohue, C. J., Holmes, D. F., & Svintsitski, O. G. (1996). A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs. Mathematical Programming, 75, 327–352.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming (Springer series in operations research). New York: Springer.
Carøe, C. C., & Tind, J. (1998). L-shaped decomposition of two-stage stochastic programs with integer recourse. Mathematical Programming, 83, 451–464.
Charnes, A., & Cooper, W. W. (1959). Chance constrained programming. Management Science, 6, 73–79.
Committee, J. E. P. S. (1987). Denryoku-jyuyou-soutei oyobi Denryoku-kyoukyuu-keikaku Santei-houshiki no Kaisetsu (in Japanese).
Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 1, 197–206.
Drezner, Z. (1992). Computation of the multivariate normal integral. ACM Transactions on Mathematical Software, 18, 470–480.
Ermoliev, Y., & Wets, R. J.-B. eds. (1988). Numerical techniques for stochastic optimization (Springer series in computational mathematics, Vol. 10). New York: Springer.
Fourer, R., Gay, D. M., & Kernighan, B. W. (1993). AMPL: A modeling language for mathematical programming. San Francisco: Scientific Press.
Graham, R. L. (1972). An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1, 132–133.
Haneveld, W. K., Stougie, L., & van der Vlerk, M. (1996). An algorithm for the construction of convex hulls in simple integer recourse programming. Annals of Operations Research, 64, 67–81.
Kall, P., & Wallace, S. W. (1994). Stochastic programming. Chichester: Wiley.
Klein Haneveld, W. K., Stougie, L., & van der Vlerk, M. H. (1995). On the convex hull of the simple integer recourse objective function. Annals of Operations Research, 56, 209–224.
Laporte, G., & Louveaux, F. V. (1993). The integer l-shaped method for stochastic integer programs with recourse. Operations Research Letters, 13, 133–142.
Louveaux, F. V. (1980). A solution method for multistage stochastic programs with recourse with application to an energy investment. Operations Research, 28, 889–902.
Louveaux, F. V. (1986). Multistage stochastic programs with block-separable recourse. Mathematical Programming Study, 28, 48–62.
Louveaux, F. V., & van der Vlerk, M. H. (1993). Stochastic programming with simple integer recourse. Mathematical Programming, 61, 301–325.
Prékopa, A. (1971). Logarithmic concave measures with application to stochastic programming. Acta Scientiarum Mathematicarum, 32, 301–316.
Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum, 34, 335–343.
Prékopa, A. (1988). Numerical solution of probabilistic constrained problems. In: Numerical Techniques for Stochastic Optimization (Yu. Ermoliev and R.J.B Wets, eds.). Springer, 1987, 123–129.
Prékopa, A. (1995). Stochastic programming. Dordrecht/Boston/London: Kluwer Academic.
Preparata, F. P., & Shamos, M. I. (1985). Computational geometry: An introduction. New York: Springer.
Ruszczyński, A., & Shapiro, A. eds. (2003). Stochastic programming (Handbooks in operations research and management science, Vol. 10). Amsterdam: Elsevier.
Shiina, T. (1999). Numerical solution technique for joint chance-constrained programming problem – An application to electric power capacity expansion. Journal of the Operations Research Society of Japan, 42, 128–140.
Shiina, T. (2000a). L–shaped decomposition method for multi-stage stochastic concentrator location problem. Journal of the Operations Research Society of Japan, 43, 317–332.
Shiina, T. (2000b). Stochastic programming model for the design of computer network (in japanese). Transactions of the Japan Society for Industrial and Applied Mathematics, 10, 37–50.
Shiina, T., & Birge, J. R. (2003). Multistage stochastic programming model for electric power capacity expansion problem. Japan Journal of Industrial and Applied Mathematics, 20, 379–397.
Shiina, T., Tagaya, Y., & Morito, S. (2007). Stochastic programming with fixed charge recourse. Journal of the Operations Research Society of Japan, 50, 299–314.
Shiina, T., Tagaya, Y., & Morito, S. (2010). Stochastic programming considering variance (in Japanese). Transactions of the Operations Research Society of Japan, 53, 114–132.
Van Slyke, R., & Wets, R. J.-B. (1969). L-shaped linear programs with applications to optimal control and stochastic linear programs. SIAM Journal on Applied Mathematics, 17, 638–663.
Wets, R. J.-B. (1989). Stochastic programming. In G. L. Nemhauser, A. H. G. R. Kan, & M. J. Todd (Eds.), Optimization (Handbooks in operations research and management science, Vol. 1, pp. 573–629). Amsterdam: Elsevier.
Yamauchi, Z. ed. (1972). STATISTICAL TABLES and formulas with computer applications. Tokyo: Japanese Standard Association.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Xu, C., Shiina, T. (2018). Basic Results on Stochastic Programming. In: Risk Management in Finance and Logistics. Translational Systems Sciences, vol 14. Springer, Singapore. https://doi.org/10.1007/978-981-13-0317-3_6
Download citation
DOI: https://doi.org/10.1007/978-981-13-0317-3_6
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-0316-6
Online ISBN: 978-981-13-0317-3
eBook Packages: Economics and FinanceEconomics and Finance (R0)