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

Discrete-choice network revenue management (DC-NRM) captures both customer behavior and the resource-usage interaction of products, and is appropriate for airline and hotel revenue management, dynamic sales of bundles in advertising, and dynamic assortment optimization in retail. The state-space of the DC-NRM stochastic dynamic program explodes and approximation methods such as the choice deterministic linear program, the affine, and the piecewise-linear approximations have been proposed to approximate it in practice. The affine relaxation (and thereby, its generalization, the piecewise-linear approximation) is intractable even for the simplest choice models such as the multinomial logit (MNL) choice model with a single segment. In this paper we propose a new Lagrangian relaxation method for DC-NRM based on an extended set of multipliers. An attractive feature of our method is that the number of constraints in our formulation scales linearly with the resource capacities. While the number of constraints in our formulation is an order of magnitude smaller that the piecewise-linear approximation (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 are small in size—a reasonable modeling tradeoff in many practical applications—our method is an indirect way to obtain the piecewise-linear approximation on large problems effectively. Our results 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 relaxation method can provide substantial improvements over existing benchmark methods, both in terms of tighter upper bounds, as well as revenues from policies based on the relaxation.