Measures of information in multi-stage stochastic programming(Bounds in Multistage Linear Stochastic Programming)

Multistage stochastic programs, which involve sequences of decisions over time, areusually hard to solve in realistically sized problems. In the two-stage case, several approaches basedon different levels of available information has been adopted in literature such as the Expected ValueProblem, 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 thequality 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 ExpectedValue of the Reference Scenario, M EV RS, the Multistage Sum of Pairs Expected Values, M SP EVand the Multistage Expectation of Pairs Expected Value, M EP EV by means of the new concept ofauxiliary scenario and redefinition of pairs subproblems probability. We show that theorems provedin [2] and [3] for two stage case are valid also in the multi-stage case. Measures of quality of theaverage solution such as the Multistage Loss Using Skeleton Solution, M LU SSt and the MultistageLoss 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. Thesebounds may help in evaluating whether it is worth the additional computations for the stochasticprogram versus the simplified approaches proposed. Numerical results on a case study related to asimple transportation problem are shown