New compact 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, and a linear program approximation called 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 has been proposed. In this paper we obtain a better approximation to the dynamic program than CDLP while retaining the appealing properties of a compact linear programming representation. Our formulation is based on 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 provably better, 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 where choice by each segment is according to the MNL model. Finally we perform extensive numerical comparisons on the various methods to evaluate their performance.