Reduced cost-based variable fixing in two-stage stochastic programming

Abstract

The explicit consideration of uncertainty is essential in addressing most planning and operation issues encountered in the management of complex systems. Unfortunately, the resulting stochastic programming formulations, integer ones in particular, are generally hard to solve when applied to realistically-sized instances. A common approach is to consider the simpler deterministic version of the formulation, even if it is well known that the solution quality could be arbitrarily bad. In this paper, we aim to identify meaningful information, which can be extracted from the solution of the deterministic problem, in order to reduce the size of the stochastic one. Focusing on two-stage formulations, we show how and under which conditions the reduced costs associated to the variables in the deterministic formulation can be used as an indicator for excluding/retaining decision variables in the stochastic model. We introduce a new measure, the Loss of Reduced Costs-based Variable Fixing (LRCVF), computed as the difference between the optimal values of the stochastic problem and its reduced version obtained by fixing a certain number of variables. We relate the LRCVF with existing measures and show how to select the set of variables to fix. We then illustrate the interest of the proposed LRCVF and related heuristic procedure, in terms of computational time reduction and accuracy in finding the optimal solution, by applying them to a wide range of problems from the literature.