The value of information in multi-stage 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, SPEV, the Expectation of Pairs Expected Value, EPEV, 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 upgradability. Chains of inequalities among the new quantities are proved to evaluate if it is worth the additional computations for the stochastic program versus the simplified approaches proposed. Numerical results on a simple transportation problem are shown. In this paper we generalize the definition of the above quantities to the multistage stochastic framework introducing the Multistage Expected Value of the Reference Scenario, MEV RS, the Multistage Sum of Pairs Expected Values, MSPEV and the Multistage Expectation of Pairs Expected Value, MEPEV by means of the new concept of auxiliary scenario and redefinition of pairs subproblems probability. Measures of quality of the average solution such as the Multistage Loss Using Skeleton Solution, MLUSSt and the Multistage Loss of Upgrading the Deterministic Solution, MLUDSt are introduced too and related to the standard Value of Stochastic Solution, V SSt at stage t.