Bounds in Multistage Linear Stochastic Programming

Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships.

[1]  James G. Morris,et al.  Stochastic Programming with Recourse: A Modification from an Applications Viewpoint , 1978 .

[2]  H. Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[3]  John R. Birge,et al.  Introduction to Stochastic programming (2nd edition), Springer verlag, New York , 2011 .

[4]  Michal Kaut,et al.  Stochastic optimization models for a single-sink transportation problem , 2009, Comput. Manag. Sci..

[5]  Marida Bertocchi,et al.  A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants , 2012, Ann. Oper. Res..

[6]  Georg Ch. Pflug.  The Value of Perfect Information as a Risk Measure , 2004 .

[7]  Shabbir Ahmed,et al.  The value of multi-stage stochastic programming in capacity planning under uncertainty , 2005 .

[8]  Georg Ch. Pflug,et al.  Dynamic Stochastic Optimization , 2004 .

[9]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[10]  Mordecai Avriel,et al.  The Value of Information and Stochastic Programming , 1970, Oper. Res..

[11]  Georg Ch. Pflug,et al.  Measuring Risk for Income Streams , 2005, Comput. Optim. Appl..

[12]  John M. Wilson,et al.  Introduction to Stochastic Programming , 1998, J. Oper. Res. Soc..

[13]  Jitka Čížková.  Expected value of information in stochastic programming , 2006 .

[14]  W. T. Ziemba,et al.  Bounds on the value of information in uncertain decision problems , 1975 .

[15]  Howard Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[16]  John R. Birge,et al.  The value of the stochastic solution in stochastic linear programs with fixed recourse , 1982, Math. Program..

[17]  Kai Huang,et al.  The Value of Multistage Stochastic Programming in Capacity Planning Under Uncertainty , 2009, Oper. Res..

[18]  Stein W. Wallace,et al.  Analyzing the quality of the expected value solution in stochastic programming , 2012, Ann. Oper. Res..

[19]  A. Madansky.  Inequalities for Stochastic Linear Programming Problems , 1960 .

[20]  R. Wets,et al.  Stochastic programming , 1989 .

[21]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[22]  Alexander Shapiro,et al.  Stochastic programming approach to optimization under uncertainty , 2007, Math. Program..

[23]  Oded Berman,et al.  Models for planning capacity expansion of convenience stores under uncertain demand and the value of information , 1995, Ann. Oper. Res..

[24]  L. Escudero,et al.  The value of the stochastic solution in multistage problems , 2007 .