New compact linear programming formulations for choice network revenue management

The choice network revenue management model incorporates customer purchase behavior as a function of the offered products, and is the appropriate model for airline and hotel network revenue management, dynamic sales of bundles, and dynamic assortment optimization. The optimization problem is a stochastic dynamic program and is intractable. A certainty-equivalence relaxation of the dynamic program, called the choice deterministic linear program (CDLP ) is usually used to generate dynamic controls. Recently a compact linear programming formulation of CDLP for the multinomial-logit (MNL) model of customer choice with non-overlapping consideration sets has been proposed. Our objective is to obtain a tighter bound than CDLP while retaining the appealing properties of a linear programming representation. To this end, it is natural to consider the affine relaxation of the dynamic program. We first show that the affine relaxation is NP-complete even for a single-segment MNL model. Nevertheless, by analyzing the affine relaxation we derive new linear programs that approximate the dynamic programming value function better than CDLP , provably between the CDLP value and the affine relaxation, and often coming close to the latter in our numerical experiments. We give extensions to the case with multiple customer segments and nested-logit model of choice. Finally we perform extensive numerical comparisons on the various bounds to evaluate their performance.