Network revenue management with inventory-sensitive bid prices and customer choice

Abstract

We develop an approximate dynamic programming approach to network revenue management models with customer choice that approximates the value function of the Markov decision process with a non-linear function which is separable across resource inventory levels. This approximation can exhibit significantly improved accuracy compared to currently available methods. It further allows for arbitrary aggregation of inventory units and thereby reduction of computational workload, yields upper bounds on the optimal expected revenue that are provably at least as tight as those obtained from previous approaches. Computational experiments for the multinomial logit choice model with distinct consideration sets show that policies derived from our approach can outperform some recently proposed alternatives, and we demonstrate how aggregation can be used to balance solution quality and runtime.

Introduction

A particular area of revenue management (RM) that currently receives much interest is the approximate solution of the RM network problem including models of customer choice behavior. Network problems arise in many applications such as hospitality or transportation where the managed products might require more than one resource, for example a hotel that sells rooms over several nights. While network models have been around for some time already, only in recent years researchers devoted themselves to advancing discrete choice models where the purchase decisions also depend on the offered product alternatives. The need for such models is heightened by the rise of low cost service providers since they cut many of the traditional restrictions meant to segment the market, leaving the customer with similar products whose essentially only distinguishing feature is the price. Even if there are still some restrictions, customers increasingly tend to ignore them in their purchase decision so that in some business applications demand can only be observed for the product with the lowest available price, as pointed out by Boyd and Kallesen (2004). Such a behavior is in stark contrast to the traditional independent demand setting where it is assumed that demand is associated with a product and does not depend on market conditions such as which other products the firm offers. Therefore it is crucial to incorporate customer choice models into RM; more on the advantages of customer choice in the RM context can be found in van Ryzin (2005) and, for a comprehensive treatment of RM, in Talluri and van Ryzin (2004b).

We base our investigations on the particularly interesting work of Zhang and Adelman (2009) who extend the previous independent demand RM model of Adelman (2007) to incorporate customer choice behavior. Their approach differs from others in that they use an affine function of the state vector to approximate the value function of the exact dynamic programming formulation with a linear program (LP) in a way such that it yields time-dependent estimates of the marginal capacity values. The optimal objective of this LP constitutes an upper bound on the exact optimal expected revenue which is tighter than those obtained by several other currently available methods. Since the LP possesses many variables, solving the problem by column generation is shown for the multinomial logit choice model (MNL) with disjoint consideration sets to reduce essentially to solving smaller mixed integer linear programs and is thus implementable in practice. They construct policies directly from the dual solution as well as through a dynamic programming decomposition scheme and show that both perform very well. The most important reason for the improved performance is that the LP naturally generates time-dependent marginal capacity value estimates which gives this approach a cutting edge compared to methods that generate static values.

However, intuitively these values should not only depend on time to departure (for the ease of presentation we will stick to airline terminology), but also on the inventory levels. This dependence on intermediate capacity levels of the resources is not captured by current approaches to network RM with choice behavior. In the independent demand setting, a suitable approximation function was recently proposed by Farias and Van Roy (2007). Instead of using constraint generation to deal with the many constraints of the arising linear program they propose using a constraint sampling procedure which is based on the work of de Farias and Van Roy, 2003, de Farias and Van Roy, 2004. The same approximation was independently proposed by Talluri (2008) under the name of strong affine relaxation and shown to provide tighter upper bounds on the optimal expected revenue than other available methods for the no-choice setting. Also Topaloglu (2009) recently focussed on time- and capacity-dependent bid prices: He proposed a network RM approach based on Lagrangian relaxation, but again without inclusion of choice behavior.

Our key contributions are the following:

• We propose a new linear programming approach to approximate dynamic programming that approximates the value function with a nonlinear function of the state vector which is separable over arbitrarily chosen ranges of resource inventory levels. As a special case, we can choose this approximation to be separable over each possible inventory level, which then corresponds to the approximation proposed by Farias and Van Roy (2007), but, in contrast to their approach, our model also accounts for customer choice behavior.

• We show that all the linear programs of Liu and van Ryzin, 2008, Zhang and Adelman, 2009, Kunnumkal and Topaloglu, 2008 can be seen as special cases of our linear programming formulation. In particular, for that reason we obtain tighter upper bounds on the objective value than these other approaches and that are asymptotically optimal as time horizon, demand and capacities are linearly scaled up.

• We prove that column generation essentially reduces to solving small mixed integer linear programs. Policies for the MNL model with disjoint consideration sets are numerically tested and show significantly improved results.

• Due to the larger number of constraints, our approach is considerably more expensive than others if we allow the marginal capacity value estimates to change from any possible inventory level to another. However, we find that sensitivity to inventory levels is most pronounced only relatively close to the departures: Therefore, in order to cut down computational requirements for large networks without much deterioration of the solution quality, we can exploit the flexibility of our model with respect to arbitrary aggregations of inventory levels to solve it with high inventory aggregation at the beginning of the booking horizon, and later to re-solve it with lower aggregation and thus higher accuracy so that we capture the typically more pronounced nonlinearity in inventory levels of the value function closer to the end of the time horizon.

• A seemingly new upper bound relationship between the approaches of Zhang and Adelman, 2009, Kunnumkal and Topaloglu, 2008 is shown, namely that the former provides a tighter upper bound on the objective value than the latter.

The paper at hand is organized as follows: In the next section we briefly review the related literature, then in Section 3 we present our model including the required notation followed by the resulting Markov decision process and its equivalent linear programming form in Section 4. We introduce the linear programming models that we compare our approach with in Sections 4.1 Choice-based deterministic LP, 4.2 Alternative deterministic LP, 5 Approximation based on the equivalent LP. Our own approach is derived in Section 5 as well. We show that the column generation subproblem is reducible to a mixed integer linear program in Section 6 and describe bid price policies in Section 7. Finally, we present the computational results in Section 8 and conclude in Section 9.