On stochastic dynamic programming for solving large-scale planning problems under uncertainty

• Authors: M. Pilar Cristobal;Laureano F. Escudero;Juan F. Monge
• Venue: Computers and Operations Research
• Year: 2009

The stochastic dynamic programming approach outlined here, makes use of the scenario tree in a back-to-front scheme. The multi-period stochastic problems, related to the subtrees whose root nodes are the starting nodes (i.e., scenario groups), are solved at each given stage along the time horizon. Each subproblem considers the effect of the stochasticity of the uncertain parameters from the periods of the given stage, by using curves that estimate the expected future value (EFV) of the objective function. Each subproblem is solved for a set of reference levels of the variables that also have nonzero elements in any of the previous stages besides the given stage. An appropriate sensitivity analysis of the objective function for each reference level of the linking variables allows us to estimate the EFV curves applicable to the scenario groups from the previous stages, until the curves for the first stage have been computed. An application of the scheme to the problem of production planning with logical constraints is presented. The aim of the problem consists of obtaining the planning of tactical production over the scenarios along the time horizon. The expected total cost is minimized to satisfy the product demand. Some computational experience is reported. The proposed approach compares favorably with a state-of-the-art optimization engine in instances on a very large scale. Scope and purpose: For quite some time, we have known that traditional methods of deterministic optimization are not suitable to capture the truly dynamic nature of most real-life problems, in view of the fact that the parameters which represent information concerning the future are uncertain. Many of the problems in planning under uncertainty, have logical constraints that require 0-1 variables in their formulation and can be solved via stochastic integer programming using scenario tree analysis. Given the dimensions of the deterministic equivalent model (DEM) of the stochastic problem, certain decomposition approaches can be considered by exploiting the structure of the models. Traditional decomposition schemes, such as the Benders and Lagrangean approaches, do not appear to provide the solution for large-scale problems (mainly in the cardinality of the scenario tree) in affordable computing effort. In this work, a stochastic dynamic programming approach is suggested, which we feel is particularly suited to exploit the scenario tree structure and, therefore, very amenable to finding solutions to very large-scale DEMs. The pilot case used involves a classical tactical production planning problem, where the structure is not exploited by the proposed approach so that it is generally applicable.