Gustavo J. Vulcano

A Column Generation Algorithm for Choice-Based Network Revenue Management

During the past few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges. One way to describe choice behavior is to assume that each customer belongs to a segment, which is characterized by a consideration set, i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature. In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. (2004) [Gallego, G., G. Iyengar, R. Phillips, A. Dubey. 2004. Managing flexible products on a network. Technical Report CORC TR-2004-01, Department of Industrial Engineering and Operations Research, Columbia University, New York], and the follow-up dynamic programming decomposition heuristic of van Ryzin and Liu (2008) [van Ryzin, G. J., Q. Liu. 2008. On the choice-based linear programming model for network revenue management. Manufacturing Service Oper. Management10(2) 288--310]. We focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-hard and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective and that the overall approach leads to high-quality, practical solutions.

Estimating Primary Demand for Substitutable Products from Sales Transaction Data

We propose a method for estimating substitute and lost demand when only sales and product availability data are observable, not all products are displayed in all periods (e.g., due to stockouts or availability controls), and the seller knows its aggregate market share. The model combines a multinomial logit (MNL) choice model with a nonhomogeneous Poisson model of arrivals over multiple periods. Our key idea is to view the problem in terms of primary (or first-choice) demand; that is, the demand that would have been observed if all products had been available in all periods. We then apply the expectation-maximization (EM) method to this model, and we treat the observed demand as an incomplete observation of primary demand. This leads to an efficient, iterative procedure for estimating the parameters of the model. All limit points of the procedure are provably stationary points of the incomplete data log-likelihood function. Every iteration of the algorithm consists of simple, closed-form calculations. We illustrate the effectiveness of the procedure on simulated data and two industry data sets.

OM Practice---Choice-Based Revenue Management: An Empirical Study of Estimation and Optimization

Discrete choice models are appealing for airline revenue management (RM) because they offer a means to profitably exploit preferences for attributes such as time of day, routing, brand, and price. They are also good at modeling demand for unrestricted fare class structures, which are widespread throughout the industry. However, there is little empirical research on the practicality and effectiveness of choice-based RM models. Toward this end, we report the results of a study of choice-based RM conducted with a major U.S. airline. Our study had two main objectives: (1) to assess the extent to which choice models can be estimated well using readily available airline data, and (2) to gauge the potential impact that choice-based RM could have on a sample of test markets. We developed a maximum likelihood estimation algorithm that uses a variation of the expectation-maximization method to account for unobservable data. The procedure was applied to data for a test market from New York City to a destination in Florida. The outputs are promising in terms of the quality of the computed estimates, although a large number of departure instances may be necessary to achieve highly accurate results. These choice model estimates were then used in a simulation study to assess the revenue performance of the EMSR-b (expected marginal seat revenue, version b) capacity control policies and the current controls used by the airline relative to controls optimized to account for choice behavior. Our simulation results show 1%--5% average revenue improvements using choice-based RM. Although such simulated results must be taken with caution, overall our study suggests that choice-based revenue management is both feasible to execute and economically significant in real-world airline environments.

Computing Virtual Nesting Controls for Network Revenue Management Under Customer Choice Behavior

We consider a revenue management, network capacity control problem in a setting where heterogeneous customers choose among the various products offered by a firm (e.g., different flight times, fare classes, and/or routings). Customers may therefore substitute if their preferred products are not offered. These individual customer choice decisions are modeled as a very general stochastic sequence of customers, each of whom has an ordered list of preferences. Minimal assumptions are made about the statistical properties of this demand sequence. We assume that the firm controls the availability of products using a virtual nesting control strategy and would like to optimize the protection levels for its virtual classes accounting for the (potentially quite complex) choice behavior of its customers. We formulate a continuous demand and capacity approximation for this problem, which allows for the partial acceptance of requests for products. The model admits an efficient calculation of the sample path gradient of the network revenue function. This gradient is then used to construct a stochastic steepest ascent algorithm. We show the algorithm converges in probability to a stationary point of the expected revenue function under mild conditions. The algorithm is relatively efficient even on large network problems, and in our simulation experiments it produces significant revenue increases relative to traditional virtual nesting methods. On a large-scale, real-world airline example using choice behavior models fit to actual booking data, the method produced an estimated 10% improvement in revenue relative to the controls used by the airline. The examples also provide interesting insights into how protection levels should be adjusted to account for choice behavior. Overall, the results indicate that choice behavior has a significant impact on both capacity control decisions and revenue performance and that our method is a viable approach for addressing the problem.

Online Auction and List Price Revenue Management

We analyze a revenue management problem in which a seller facing a Poisson arrival stream of consumers operates an online multiunit auction. Consumers can get the product from an alternative list price channel. We consider two variants of this problem: In the first variant, the list price is an external channel run by another firm. In the second one, the seller manages both the auction and the list price channels. Each consumer, trying to maximize his own surplus, must decide either to buy at the posted price and get the item at no risk, or to join the auction and wait until its end, when the winners are revealed and the auction price is disclosed. Our approach consists of two parts. First, we study structural properties of the problem, and show that the equilibrium strategy for both versions of this game is of the threshold type, meaning that a consumer will join the auction only if his arrival time is above a function of his own valuation. This consumers strategy can be computed using an iterative algorithm in a function space, provably convergent under some conditions. Unfortunately, this procedure is computationally intensive. Second, and to overcome this limitation, we formulate an asymptotic version of the problem, in which the demand rate and the initial number of units grow proportionally large. We obtain a simple closed-form expression for the equilibrium strategy in this regime, which is then used as an approximate solution to the original problem. Numerical computations show that this heuristic is very accurate. The asymptotic solution culminates in simple and precise recipes of how bidders should behave, as well as how the seller should structure the auction, and price the product in the dual-channel case.

Simulation-Based Optimization of Virtual Nesting Controls for Network Revenue Management

Virtual nesting is a popular capacity control strategy in network revenue management. In virtual nesting, products (itinerary-fare-class combinations) are mapped (“indexed”) into a relatively small number of “virtual classes” on each resource (flight leg) of the network. Nested protection levels are then used to control the availability of these virtual classes; specifically, a product request is accepted if and only if its corresponding virtual class is available on each resource required. Bertsimas and de Boer proposed an innovative simulation-based optimization method for computing protection levels in a virtual nesting control scheme [Bertsimas, D., S. de Boer. 2005. Simulation-based booking-limits for airline revenue management. Oper. Res.53 90--106]. In contrast to traditional heuristic methods, this simulation approach captures the true network revenues generated by virtual nesting controls. However, because it is based on a discrete model of capacity and demand, the method has both computational and theoretical limitations. In particular, it uses first-difference estimates, which are computationally complex to calculate exactly. These gradient estimates are then used in a steepest-ascent-type algorithm, which, for discrete problems, has no guarantee of convergence. In this paper, we analyze a continuous model of the problem that retains most of the desirable features of the Bertsimas-de Boer method, yet avoids many of its pitfalls. Because our model is continuous, we are able to compute gradients exactly using a simple and efficient recursion. Indeed, our gradient estimates are often an order of magnitude faster to compute than first-difference estimates, which is an important practical feature given that simulation-based optimization is computationally intensive. In addition, because our model results in a smooth optimization problem, we are able to prove that stochastic gradient methods are at least locally convergent. On several test problems using realistic networks, the method is fast and produces significant performance improvements relative to the protection levels produced by heuristic virtual nesting schemes. These results suggest it has good practical potential.

Revenue Sharing in Airline Alliances

We propose a two-stage game-theoretic approach to study the operations of an airline alliance in which independent carriers, managing different reservation and information systems, can collaboratively market and operate codeshare and interline itineraries. In the first-stage game, airlines negotiate fixed proration rates to share the revenues generated by such itineraries. In the second-stage game, airlines operate independent inventory control systems to maximize their own expected revenues. We derive a revenue-sharing rule that is (i) an admissible outcome of the first-stage negotiation, in the sense that no airline coalition has enough incentives to secede from the grand alliance, and (ii) efficient for the second-stage game, in the sense that the decentralized system can achieve the same revenues as a central planner managing the global alliance network. Our numerical study shows that the proposed proration rates can lead to a significant increase in revenues with respect to other rules commonly used in practice. Finally, because our proposal requires the disclosure of private demand information, we introduce a simple alternative rule that is based on public information. This heuristic performs remarkably well, becoming an interesting candidate to be pursued in practice. This paper was accepted by Martin Lariviere, operations management.

Computing Bid Prices for Revenue Management Under Customer Choice Behavior

We consider a choice-based, network revenue management (RM) problem in a setting where heterogeneous customers consider an assortment of products offered by a firm (e.g., different flight times, fare classes, and/or routes). Individual choice decisions are modeled through an ordered list of preferences, and minimal assumptions are made about the statistical properties of this demand sequence. The firm manages the availability of products using a bid-price control strategy, and would like to optimize the control parameters. We formulate a continuous demand and capacity model for this problem that allows for the partial acceptance of requests. The model admits a simple calculation of the sample path gradient of the revenue function. This gradient is then used to construct a stochastic steepest ascent algorithm. We show that the algorithm converges (w.p.1) to a stationary point of the expected revenue function under mild conditions. The procedure is relatively efficient from a computational standpoint, and in our synthetic and real-data experiments performs comparably to or even better than other choice-based methods that are incompatible with the current infrastructure of RM systems. These features make it an interesting candidate to be pursued for real-world applications.

A branch-and-cut algorithm for the latent-class logit assortment problem

We study the product assortment problem of a retail operation that faces a stream of customers who are heterogeneous with respect to preferences. Each customer belongs to a market segment characterized by a consideration set that includes the alternatives viewed as options, and by the preference weights that the segment assigns to each of those alternatives. Upon arrival, he checks the offer set displayed by the firm, and either chooses one of those products or quits without purchasing according to a multinomial-logit (MNL) criterion. The firms goal is to maximize the expected revenue extracted during a fixed time horizon. This problem also arises in the growing area of choice-based, network revenue management, where computational speed is a critical factor for the practical viability of a solution approach. This so-called latent-class, logit assortment problem is known to be NP-Hard. In this paper, we analyze unconstrained and constrained (i.e., with a limited number of products to display) versions of it, and propose a branch-and-cut algorithm that is computationally fast and leads to (nearly) optimal solutions.

A Market Discovery Algorithm to Estimate a General Class of Nonparametric Choice Models

We propose an approach for estimating customer preferences for a set of substitutable products using only sales transactions and product availability data. The underlying demand framework combines a general, nonparametric discrete choice model with a Bernoulli process of arrivals over time. The choice model is defined by a discrete probability mass function pmf on a set of possible preference rankings of alternatives, and it is compatible with any random utility model. An arriving customer is assumed to purchase the available option that ranks highest in her preference list. The problem we address is how to jointly estimate the arrival rate and the pmf of the rank-based choice model under a maximum likelihood criterion. Since the potential number of customer types is factorial, we propose a market discovery algorithm that starts with a parsimonious set of types and enlarge it by automatically generating new types that increase the likelihood value. Numerical experiments confirm the potential of our proposal. For a realistic data set in the hospitality industry, our approach improves the root mean square errors between predicted and observed purchases computed under independent demand model estimates by 67% to 93%. This paper was accepted by Serguei Netessine, operations management.