Vol. 43, No. 3, August 2009, pp. 381–394 issn 0041-1655  eissn 1526-5447  09  4303  0381 inf orms ® doi 10.1287/trsc.1090.0262 © 2009 INFORMS An Approximate Dynamic Programming Approach to Network Revenue Management with Customer Choice Dan Zhang Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, Canada, dan.zhang@mcgill.ca Daniel Adelman Graduate School of Business, University of Chicago, Chicago, Illinois 60637, dan.adelman@chicagogsb.edu W e consider a network revenue management problem where customers choose among open fare products according to some prespecified choice model. Starting with a Markov decision process (MDP) formulation, we approximate the value function with an affine function of the state vector. We show that the resulting problem provides a tighter bound for the MDP value than the choice-based linear program. We develop a column generation algorithm to solve the problem for a multinomial logit choice model with disjoint consideration sets (MNLD). We also derive a bound as a by-product of a decomposition heuristic. Our numerical study shows the policies from our solution approach can significantly outperform heuristics from the choice-based linear program. Key words : network revenue management; choice behavior; dynamic programming History : Received: August 2006; revisions received: October 2007 and June 2008; accepted: December 21, 2008. Published online in Articles in Advance June 29, 2009. While substantial research has been done on meth- ods for solving the network revenue management problem, much less work has been done in solving the version where customers choose among avail- able network products. Usually, when airlines open up a menu of fares for a given set of flights, cus- tomers will make substitutions between those avail- able, or purchase nothing. Although incorporating customer choice is important in practice, methodolog- ically is more difficult than the independent demand case, which already suffers from Bellman’s “curse of dimensionality.” Recently, Liu and van Ryzin (2008) studied a lin- ear programming formulation, which they call the choice-based linear program. In essence, their linear pro- gramming formulation is the choice equivalent of the widely used deterministic linear program (DLP) for network revenue management. It approximates the original stochastic problem by replacing stochastic demand with its expected value. They provide a col- umn generation algorithm to solve the problem for the multinomial logit choice model with disjoint consid- eration sets (MNLD). The linear programming formu- lation is the same as the model proposed in Gallego et al. (2004), where the focus is on analyzing so-called flexible products. The purpose of this paper is to extend the approx- imate dynamic programming approach of Adelman (2007) to the customer choice setting, and compare it to Liu and van Ryzin (2008). This is an emerging approach to a wide variety of problems in operations research, for which a lot of active research is ongoing (see, e.g., de Farias and Van Roy 2003). The idea is to formulate the underlying dynamic program as a linear program, and then make an affine functional approx- imation to the value function to obtain dynamic bid- prices in which marginal resource values change as a function of time. In the independent demand setting, these dynamic bid-prices perform better, in terms of both the bound and policy obtained, than the static bid-prices obtained from the standard linear program. We discover in this paper that this statement remains true when the method is extended to the choice set- ting. In fact, the gap between the bounds obtained empirically can be as much as 50%. In addition to the parallel results to Adelman (2007), there are two unique contributions in this paper. First, we provide a way to solve the column generation sub- problem. Unlike in Adelman (2007), the column gen- eration subproblem for solving our linear program is a nontrivial nonlinear integer programming problem for general discrete choice models. For the MNLD choice model, the subproblem belongs to the class of integer generalized fractional programs. General algorithms for efficient solution of such problems are not currently 381 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/.