2012-03-19Buch DOI: 10.18452/3047
Measures of information in multi-stage stochastic programming(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. In the two-stage case, several approaches based on different levels of available information has been adopted in literature such as the Expected Value Problem, EV , the Sum of Pairs Expected Values, SP EV , the Expectation of Pairs Expected Value, EP EV, solving series of sub-problems more computationally tractable than the initial one, or the Expected Skeleton Solution Value, ESSV and the Expected Input Value, EIV which evaluate the quality of the deterministic solution in term of its structure and upgradeability. In this paper we generalize the deﬁnition of the above quantities to the multistage stochastic for- mulation when the right hand side of constraints are stochastic: we introduce the Multistage Expected Value of the Reference Scenario, M EV RS, the Multistage Sum of Pairs Expected Values, M SP EV and the Multistage Expectation of Pairs Expected Value, M EP EV by means of the new concept of auxiliary scenario and redeﬁnition of pairs subproblems probability. We show that theorems proved in  and  for two stage case are valid also in the multi-stage case. Measures of quality of the average solution such as the Multistage Loss Using Skeleton Solution, M LU SSt and the Multistage Loss of Upgrading the Deterministic Solution, M LU DSt are introduced and related to the standard Value of Stochastic Solution, V SSt at stage t. A set of theorems providing chains of inequalities among the new quantities are proved. These bounds may help in evaluating whether it is worth the additional computations for the stochastic program versus the simpliﬁed approaches proposed. Numerical results on a case study related to a simple transportation problem are shown.
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