A note on relaxations of the choice network revenue management dynamic program

In recent years, several approximation methods have been proposed for the choice network revenue management problem. These approximation methods are proposed because the dynamic programming formulation of the choice network revenue management problem is intractable even for moderately sized instances. In this paper, we consider three approximation methods that obtain upper bounds on the value function, namely, the choice deterministic linear program (CDLP), the affine approximation (AF), and the piecewise-linear approximation (PL). It is known that the piecewise-linear approximation bound is tighter than the affine bound, which in turn is tighter than CDLP. In this paper, we prove bounds on how much the affine and piecewise-linear approximations can tighten CDLP. We show (i) the gap between the AF and CDLP bounds is at most a factor of 1+1/(mini{r1i}), where r1i>0 are the resource capacities, and (ii) the gap between the piecewise-linear and CDLP bounds is within a factor of 2. Moreover, we show that these gaps are essentially tight. Our results hold for any discrete-choice model and do not involve any asymptotic scaling. Our results are surprising because calculating the AF bound is NP-hard and CDLP is tractable for a single-segment multinomial logit model; our result implies that if a firm has all resource capacities of 100, the gap between the two bounds, however, is at most 1.01

Choice network revenue management based on new tractable approximations

The choice network revenue management model incorporates customer purchase behavior as probability of purchase as a function of the offered products, and is appropriate for air- line and hotel network revenue management, dynamic sales of bundles, and dynamic assort- ment optimization. The optimization problem is a stochastic dynamic program and is in- tractable. Consequently, a linear programming approximation called choice deterministic linear program ( CDLP ) is usually used to generate controls. Tighter approximations such as affine and piecewise-linear relaxations have been proposed, but it was not known if they can be solved efficiently even for simple models such as the multinomial logit (MNL) model with a single segment. We first show that the affine relaxation (and hence the piecewise-linear relaxation) is NP-hard even for a single-segment MNL choice model. By analyzing the affine relaxation we derive a new linear programming approximation that admits a compact representation, implying tractability, and prove that its value falls between the CDLP valueandtheaffinerelaxation value. This is the first tractable relaxation for the choice network revenue management problem that is provably tighter than CDLP . This approximation in turn leads to new policies that, in our numerical experiments, show very good promise: a 2% increase in revenue on average over CDLP ; and the values typically coming very close to the affine relaxation. We extend our analysis to obtain other tractable approximations that yield even tighter bounds. We also give extensions to the case with multiple customer segments with overlapping consideration sets where choice by each segment is according to the MNL model

A strong lagrangian relaxation for general discrete-choice network revenue management

Discrete-choice network revenue management (DC-NRM) captures both customer behaviorand the resource-usage interaction of products, and is appropriate for airline and hotel revenuemanagement, dynamic sales of bundles in advertising, and dynamic assortment optimizationin retail. The state-space of the DC-NRM stochastic dynamic program explodes and approxi-mation methods such as the choice deterministic linear program (CDLP), the affine, and thepiecewise-linear approximations have been proposed to approximate it in practice. The affinerelaxation (and thereby, its generalization, the piecewise-linear approximation) is intractableeven for the simplest choice models such as the multinomial logit (MNL) choice model with asingle segment. In this paper we propose a new Lagrangian relaxation method for DC-NRMbased on an extended set of multipliers. An attractive feature of our method is that the numberof constraints in our formulation scales linearly with the resource capacities. While the num-ber of constraints in our formulation is an order of magnitude smaller that the piecewise-linearapproximation (polynomial vs exponential), it obtains a bound that is as tight as the piecewise-linear bound. If we assume that the consideration sets of the different customer segments aresmall in size—a reasonable modeling tradeoff in many practical applications—our method is anindirect way to obtain the piecewise-linear approximation on large problems effectively. Ourresults are not specific to a particular functional form (such as MNL), but hold for any discrete-choice model of demand. We show by numerical experiments that our Lagrangian relaxationmethod can provide substantial improvements over existing benchmark methods, both in termsof tighter upper bounds, as well as revenues from policies based on the relaxation

On a piecewise-linear approximation for network revenue management

The network revenue management (RM) problem arises in airline, hotel, media, and other industries where the sale products use multiple resources. It can be formulated as a stochastic dynamic program, but the dynamic program is computationally intractable because of an exponentially large state space, and a number of heuristics have been proposed to approximate its value function. In this paper we show that the piecewise-linear approximation to the network RM dynamic program is tractable; specifically we show that the separation problem of the approximation can be solved as a relatively compact linear program. Moreover, the resulting compact formulation of the approximate dynamic program turns out to be exactly equivalent to the Lagrangian relaxation of the dynamic program, an earlier heuristic method proposed for the same problem. We perform a numerical comparison of solving the problem by generating separating cuts or as our compact linear program. We discuss extensions to versions of the network RM problem with overbooking as well as the difficulties of extending it to the choice model of network revenue RM