Abstract

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.

Highlights

  • Introduction and literature reviewIn industries such as hotels, advertising, and airlines, the products consume one multiple resources and each product has a specific price, set based on the price sensitivity of the customer segment it is aimed at, the market conditions, and the product characteristics

  • Since we work with general discrete-choice models, all the negative complexity results for the multinomial logit (MNL) model carry over for the Lagrangian relaxation as well as we show that our Lagrangian relaxation is equivalent to the piecewise-linear relaxation

  • We show how our ideas lead to a tractable method for discrete-choice Network revenue management (NRM) (DC-NRM) when the consideration sets of the different segments are small in size

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Summary

Introduction and literature review

In industries such as hotels, advertising, and airlines, the products consume one multiple resources (for example, multi-night stays, bundles of advertising timeslots, multi-leg itineraries) and each product has a specific price, set based on the price sensitivity of the customer segment it is aimed at, the market conditions, and the product characteristics. The biggest practical impact of our work is in showing that the complexity of the Lagrangian relaxation method (in terms of the number of constraints in the linear programming formulation of the problem) scales linearly with the resource capacities, while that of the piecewise-linear approximation is exponential—yet they arrive at the same value function approximation. In parallel to a working version of this paper, Vossen and Zhang [25] study the properties of approximate linear programming methods for NRM and show that the affine and piecewise-linear formulations can be significantly reduced in size. The number of constraints in the Lagrangian relaxation scales linearly with the resource capacities, while the piecewise-linear approximation is exponential, leading to a substantial reduction in running time in practice This result implies that the Lagrangian relaxation bound is stronger than the affine approximation bound ( CDLP) that was not known previously.

Problem formulation
Demand model
DC‐NRM dynamic program
Linear programming formulation of the DC‐NRM dynamic program
DP min
Value function approximation methods
Affine approximation
Piecewise‐linear approximation
Lagrangian relaxation using offer‐set specific multipliers
Preliminaries
Proof of Proposition 2
Segment‐based Lagrangian relaxation
Demand model with multiple customer segments
Tractable Lagrangian relaxation
Computational experiments
MNL choice model with multiple customer segments
Exponomial choice model with multiple customer segments
Benchmark methods
Hub‐and‐spoke network
Findings
Conclusions
Full Text
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