Bounds in Multistage Linear Stochastic Programming

Abstract

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

A scenario-based framework for supply planning under uncertainty: stochastic programming versus robust optimization approaches

“…Before to proceed with the comparison between SP and RO models, we analyze the advantage of having the information about future demand and buying cost. This is provided by the well-known Expected Value of Perfect Information EVPI, [14], [37]: EV P I = SP − W S = 107 244.67 − 84 472.21 = 22 772. 46 (92) given by the difference between the stochastic cost and the wait-and-see cost W S. Figure 2 shows the objective function values of the deterministic problems solved separately over each scenario (d s ,b s ) ∈∆ ×B.…”

Bounds and Approximations for Multistage Stochastic Programs

“…Pairs of scenarios were considered for the first time in [2], in [3] and [18] for the two-stage linear case, and in [11] for the multistage linear case by means of the definition of multistage sum of pairs expected values. Inequality (2.5) has been proven in [11] for the multistage linear case with risk functional being the expectation.…”

“…The idea of looking at subproblems with only two scenarios goes back to [2] for the two-stage linear case (see also [3] and [18]). An extension to the multistage linear case is in [11].…”

“…Lower bounds for replacing the scenario process by its expectation are also considered. Several upper bounds obtained by inserting feasible solutions derived from smaller subproblems are presented generalizing the multistage expected value of the reference scenario (MEVRS) and the multistage expectation of pairs expected value (MEPEV) introduced in [11]. Relations among them are derived.…”

The Multi-path Traveling Salesman Problem with Stochastic Travel Costs: Building Realistic Instances for City Logistics Applications

“…In these problems, a known distribution affecting some problem parameters is given and the theoretical results are strongly connected with the hypotheses on such distribution. The main sources of uncertainty are related to the arc costs, the arc location, and the subset of cities to be visited (Goemans et al, 1991;Maggioni et al, 2014;Maggioni and Wallace, 2012;Mu et al 2011;Bertazzi, & Maggioni, 2014;. Only a little set of papers deal with the choice among multiple paths and they do that by considering the shortest path problem (Eiger at al., 1985;Fu, & Rilett;1998;Psaraftis, & Tsitsiklis, 1993).…”