Pricing substitutable flights in airline revenue management

doi:10.1016/j.ejor.2006.10.067

European Journal of Operational Research

Volume 197, Issue 3, 16 September 2009, Pages 848-861

https://doi.org/10.1016/j.ejor.2006.10.067 Get rights and content

Abstract

We develop a Markov decision process formulation of a dynamic pricing problem for multiple substitutable flights between the same origin and destination, taking into account customer choice among the flights. The model is rendered computationally intractable for exact solution by its multi-dimensional state and action spaces, so we develop and analyze various bounds and heuristics. We first describe three related models, each based on some form of pooling, and introduce heuristics suggested by these models. We also develop separable bounds for the value function which are used to construct value- and policy-approximation heuristics. Extensive numerical experiments show the value- and policy-approximation approaches to work well across a wide range of problem parameters, and to outperform the pooling-based heuristics in most cases. The methods are applicable even for large problems, and are potentially useful for practical applications.

Introduction

The development of Internet distribution channels has helped create both opportunities and challenges in airline-ticket pricing. On one hand, it has allowed price changes to be made quickly and frequently with negligible costs. On the other, prices have become more visible to consumers because comparison shopping can be done with the click of a mouse. Old revenue management models that rely on the notion of “exogenous demand for a fare class” are becoming less appropriate, and consequently, it is important to develop operational models that incorporate customer choice.

From a customer’s viewpoint, flight schedule and price information is often readily available when making a purchase decision. For example, on November 22, 2006, the website of JetBlue Airways showed nine different flights spread throughout the day from 6:15 a.m. to 9:15 p.m. for a one-way trip on December 4, 2006 from New York (JFK) to Orlando (MCO). Prices differed across the flights. The earliest and latest flights were priced at $79. One of the other seven flights was priced at $124, and all the other flights were priced at either $79 or $99. Given such information on flight schedule and the price quotes, customers make their purchase decisions based on their own preferences: Do they mind taking a very early flight, or taking an evening flight and arriving late in the night? From our own experience as consumers, it is not hard to imagine that prices on the mid-day flights would affect choices of customers and effectively change demand for morning or evening flights. Given that consumers do typically choose among alternatives, how should an airline price its flights? This paper contains models to help answer this question.

We study a dynamic pricing problem for multiple substitutable flights between the same origin and destination, in which the airline’s objective is to maximize the total expected revenue from customer bookings over the finite selling horizon by setting the prices of the flights. Customers choose among the flights (or decide not to purchase) based upon their own preferences and the prices of all the flights offered. The problem we consider is particularly relevant to low-cost airlines that sell many tickets on the Internet, fly point-to-point between select city pairs, and have multiple-flights scheduled for the same day between each pair. Low-cost airlines also typically use a simplified fare structure (in comparison to those employed by traditional carriers). At each point in the booking horizon, prices for each flight are visible to customers. In addition, the effect of capacity inflexibility is even more pronounced for many low-cost airlines in comparison to traditional carriers. For example, JetBlue owns only one type of airplane (Airbus A320), which makes capacity adjustments by contingent fleet assignment impossible. Given the inflexible capacity and the simplified fare structure, pricing is a crucial lever for matching demand and capacity. Other low-fare airlines that face these issues include European carriers such as Ryanair and EasyJet. Although Internet sales make up a smaller fraction of total business for major carriers (such as Northwest, Delta, etc.), they too face problems in which customers have a choice among multiple flights between a common origin and destination. More generally, our models are applicable to retailers that employ dynamic pricing strategies to sell substitutable products over a finite horizon.

Most models for airline pricing consider one-dimensional problems with a single flight, where customer purchase decisions are based only upon the price for that flight. Such one-dimensional problems do not account for the effects of consumer behavior that are captured in our model. In our multiple-flight setting, the fact that bookings for any particular flight are influenced by the price of tickets on all the flights makes solving the problem considerably more difficult than solving multiple single-dimensional problems each with just one flight and no consumer substitution effects.

Dynamic pricing problems have been studied extensively in the economics, marketing, and operations literature. We review pricing research only in the revenue management context that is directly related to our model. In this body of literature, capacity (or inventory) is assumed to be fixed, or is prohibitively expensive to change during the selling horizon. For reviews of pricing models for revenue management, please refer to Bitran and Caldentey, 2003, Talluri and van Ryzin, 2004b, and for a survey of the literature that considers both pricing and inventory decisions, see Elmaghraby and Keskinocak (2003). We also briefly describe some revenue management models where customer choice is modeled explicitly, although no pricing decisions are involved. We close by reviewing several studies aimed at deriving customer choice parameters in the airline context.

Dynamic pricing problems for a fixed stock of a single item sold in a finite selling horizon have attracted considerable attention in the revenue management literature. Gallego and van Ryzin (1994) formulate an intensity control model of the problem and derive several structural properties. They also study a heuristic policy based on a deterministic upper bound and prove that it is asymptotically optimal. Zhao and Zheng (2000) consider a similar problem with nonhomogeneous demand and show that dynamic pricing policies can have a significant impact on revenue when demand is nonhomogeneous. Bitran and Mondschein (1997) present a continuous-time model in the context of fashion retailing and compare it to a model with periodic pricing review, where price is allowed to change only at several pre-specified time points. They show that the loss in expected revenue from implementing an appropriate periodic pricing review policy is small.

Gallego and van Ryzin (1997) consider dynamic pricing problems where a set of resources is used to produce a set of products. They develop asymptotically optimal heuristic policies and apply their results to network revenue management problems. Kleywegt (2001) considers a deterministic optimal-control formulation of a pricing problem where multiple products are sold to multiple customer classes over time. Lin and Li (2004) develop bounds on the value function of a dynamic pricing problem for a line of substitutable products. Liu and Milner (2006) study a multi-item pricing problem with a common pricing constraint. They obtain an optimal policy for a deterministic version of the problem and propose heuristics for the stochastic version.

Talluri and van Ryzin (2004a) study a single-leg revenue management problem where customers choose among the open fare classes. They prove structural properties that greatly simplify the computation of an optimal policy. Maglaras and Meissner (2006) consider the pricing problem faced by a firm that owns a fixed stock of a resource, which is used to produce several different products. Customer choice among the products is modeled by joint price elasticity. They prove structural properties that reduce the decision problem to an equivalent one-dimensional problem, and propose several heuristic policies. Iyengar et al., 2004, van Ryzin and Liu, 2007 consider choice-based linear programming models for network revenue management. van Ryzin and Vulcano (2006) consider a network revenue management problem with customer choice behavior, and propose a simulation-based optimization approach to obtain virtual nesting controls. Boyd and Kallesen (2004) discuss the impact of consumer purchase behavior on revenue management practice, distinguishing between two types of demand: yieldable, where demand is class-specific, and priceable, where demand is price-sensitive and not class-specific.

McFadden (2000) reviews the economics literature dealing with models and estimation for consumer choice in travel. Perhaps the most widely used choice models in practice are discrete-choice models (see, e.g., Ben-Akiva and Lerman, 1985). Train (2003) summarizes recent advances in discrete-choice theory and its applications, and discusses simulation-based methods to estimate choice probabilities for several discrete-choice models. Utility maximization is frequently used as a basis for deriving customer choice probabilities. Mahajan and van Ryzin (2001) point out that a number of choice models can be viewed as special cases of the utility maximization model.

Several recent works also focus on describing or fitting particular choice models in the revenue management context. Carrier (2003) considers how to model passenger preference on flight schedule, and reports results from an extensive simulation study. Algers and Beser (2001) describe how to estimate customer choice probabilities for flights and fare classes using revealed preference and stated preference data. Andersson (1998) reports on a study of passenger choice in the context of seat inventory control. Talluri and van Ryzin (2004a) use a maximum likelihood method to estimate multinomial logit choice probabilities for fare classes on a single flight.

We pose the joint pricing problem for multiple substitutable flights between the same origin and destination as a Markov decision process (MDP). The MDP has multi-dimensional state and action spaces, and therefore suffers from the well-known curse of dimensionality. Since the problem is intractable, we develop and analyze a variety of bounds and heuristics. We begin by formulating three related problems, each based upon some notion of pooling. In addition to yielding bounds on the value function of the original problem, these “pooled problems” suggest various baseline pricing heuristics for the original problem. Among these are single-price policies that, for each time period, quote the same price for all open flights (the price is, however, updated as time progresses).

We also derive separable upper and lower bounds for the value function of the original problem. These separable bounds are based upon solutions of several corresponding one-dimensional MDPs. The bounds and the associated one-dimensional problems suggest two other families of heuristics, which we term value approximation and policy approximation. These heuristics have the advantage that they remain computationally tractable, even for very large problems with many flights.

Our numerical studies show that the value- and policy-approximation heuristics appear to work well, and to perform better than the pooling-based heuristics, especially when there is asymmetry among the flights in terms of demand load and customer preferences. Several other insights also emerge from the study. For instance, the results show that the revenue loss from instituting a single-price policy can be quite significant, even when the best possible single-price policy is used. This shortcoming of single-price policies underscores the importance of using sophisticated policies that allow different prices for different flights. Moreover, this observation potentially has relevance beyond airline revenue management. In fashion retailing, as described in Bitran et al. (1998), it may be required that products at different physical locations be priced identically. Our study indicates that such a requirement (typically made to allow simpler centralized pricing control or to protect against loss of customer good will) may result in a significant loss in revenue.

We also examine via numerical experiments policies that change prices only at pre-specified time points. The analysis reveals that for practically implementable choices of such time points, the revenue loss from these policies in comparison to an optimal policy is small. The main insight here is that much of the benefit from using sophisticated dynamic pricing policies can be obtained even if the airline does not exercise complete real-time control of prices. This observation has practical significance, since airlines may want (or be able) to change prices only at certain pre-specified times, such as after daily or semi-daily database updates.

Before proceeding, we compare this paper to Zhang and Cooper (2005), hereafter ZC, which considers a seat availability problem with multiple flights between the same origin and destination in which customers choose among the open flights. For some comments comparing pricing control and availability control, see pp. 176–177 of Talluri and van Ryzin (2004b), where it is argued that pricing, when possible, is the better option. ZC assumes that customers belong to different classes that arrive sequentially in distinct periods, and that the order of the classes is pre-determined. The fare for each class is the same on all the flights. The decisions involve the number of seats to open on each flight in each period. The problem in the present paper is related to that in ZC, but differs in the following aspects. First, the present paper considers pricing decisions rather than availability decisions, and prices for different flights can be different in the same period. Second, the present paper assumes that there is at most one arrival in each period as opposed to the sequential “block-demand” setup. Third, the present paper does not assume a pre-determined order of arrivals.

The methodological approach in the present work is also related to that in ZC. Both papers consider solution procedures that bound value functions of high-dimensional MDPs with sums of value functions of one-dimensional MDPs, and both consider value-approximation heuristics. However, to apply the bounds and value-approximation procedures one must identify and exploit the structure of the specific problem. Hence, the particulars of the methods are different in the two papers. Moreover, the increased complexity of the pricing problem expands the scope of methods one might use. For instance, policy approximation does not have a counterpart in ZC, and questions regarding the frequency of price changes are not easily incorporated into the block-demand setup of our earlier work. In addition, the added complexity of the pricing problem leads us to consider three different pooling procedures in this paper, while only one form is considered in ZC.

To summarize, our main contributions are (1) formulation of the pricing problem for multiple flights with customer choice among the flights, (2) development of bounds for the value function of the MDP, (3) proposal of heuristic approaches to the problem, and (4) numerical testing that provides managerial insights and shows the proposed approaches to be potentially viable. The remainder of the paper is organized as follows. Section 2 reviews the single-flight case. Section 3 formulates the multi-flight problem. Section 4 considers various pooling models. Section 5 develops the separable bounds. Section 6 introduces the value- and policy-approximation heuristics. Section 7 reports numerical results. Section 8 provides a brief summary. Proofs are in the Appendix.

Section snippets

Preliminaries: pricing a single flight

In this section, we formulate a single-flight pricing problem as an MDP with one-dimensional state and action spaces. The model, which is a discrete-time analog of that in Zhao and Zheng (2000), is a building block for the multiple-flight case, and its solutions are, in part, the basis for heuristics we develop for the multi-flight case.

Consider a single-leg flight with capacity q. The booking horizon is divided into $τ$ discrete-time periods. The earliest period is period $τ$ , and the last period

Model formulation

In this section we formulate the multi-flight pricing problem that is the main focus of this paper. There are n flights between a single origin–destination pair. The booking horizon is divided into $τ$ discrete-time periods, and time is counted backwards. To simplify notation, we reserve the symbols i and t for flights and times, respectively, where $i \in {1, \dots, n}$ and $t \in {1, \dots, τ}$ . The capacity of flight i is $c^{i}$ . Let $c = (c^{1}, \dots, c^{n})$ . Customer arrivals are independent across time periods. In period t, there

Pooling

The MDP of the previous section has n-dimensional state and action spaces, making it intractable. In this section, we consider pooling models with one-dimensional action space, state space, or both. For models with price pooling, we assume the set of allowable prices is the same for all flights. We denote this set by $R_{0}$ , so $R_{i} = R_{0}$ for all i.

We first consider a model where at each time point a common price is quoted on all the flights with positive remaining capacity. To specify the price pooling

Separable bounds

In this section, we provide separable upper and lower bounds for $v_{t} (s)$ . The bounds are composed of value functions of several one-dimensional problems as described in Section 2. These bounds provide ingredients for computational approaches that take advantage of the relatively simple and nicely-structured solutions of one-dimensional problems.

For all i and t, let ${\underset{̲}{P}}_{t}^{i} (\cdot)$ and ${\bar{P}}_{t}^{i} (\cdot)$ be functions from $R_{i}$ to $[0, 1]$ that satisfy ${\underset{̲}{P}}_{t}^{i} (r^{i}) ⩽ P_{t}^{i} (r) ⩽ {\bar{P}}_{t}^{i} (r^{i}) \forall r$ . In Section 7, we explain how to determine

Value and policy approximation

Motivated by the analysis of the previous section, we next discuss various computationally feasible heuristic approaches for the multi-dimensional pricing problem of Section 3.

Let $β \in [0, 1]$ . In value approximation, we approximate the value function $v_{t} (s)$ and choice probabilities ${P_{t}^{i} (r)}$ by, respectively ${\tilde{v}}_{t} (s) = \sum_{i = 1}^{n} [β {\bar{v}}_{t}^{i} (s^{i}) + (1 - β) {\underset{̲}{v}}_{t}^{i} (s^{i})] and$ ${\tilde{P}}_{t}^{i} (r^{i}) = β {\bar{P}}_{t}^{i} (r^{i}) + (1 - β) {\underset{̲}{P}}_{t}^{i} (r^{i}) \forall i .$ From (8), it follows that $Δ_{i} {\tilde{v}}_{t} (s) = {\tilde{v}}_{t} (s) - {\tilde{v}}_{t} (s - ϵ^{i}) = β Δ {\bar{v}}_{t}^{i} (s^{i}) + [1 - β] Δ {\underset{̲}{v}}_{t}^{i} (s^{i})$ for all i. Hence, the approximate marginal

Numerical experiments

In this section we describe our numerical study. To show the tightness of the bounds and the effectiveness of the heuristics, we test them on examples with two flights, and compare the simulated values with exact MDP values. We also consider examples with six and 12 flights to examine how the methods perform on relatively large problems. The price set is {$150, $200, $250, $300, ρ₀} for each flight in all examples, where $ρ_{0}$ is the null price. Each flight has 80 seats unless noted otherwise. PA and

Summary

We developed a pricing model for substitutable flights where customers choose among the available flights. To overcome computational problems posed by the formulation’s multi-dimensional state and action spaces, we considered heuristics based on pooling ideas. We also derived easily-computable separable bounds for the value function of our model. Policies motivated by these bounds were shown numerically to be near optimal for a range of problem instances, and to dominate the policies from

Acknowledgment

This material is based upon work supported by the National Science Foundation under Grant Number DMI-0115385.

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View full text

Pricing substitutable flights in airline revenue management

Abstract

Introduction

Section snippets

Preliminaries: pricing a single flight

Model formulation

Pooling

Separable bounds

Value and policy approximation

Numerical experiments

Summary

Acknowledgment

International Transactions in Operational Research

Modelling choice of flight and booking class – A study using stated preference and revealed preference data

International Journal of Services Technology and Management

Discrete Choice Analysis

Neuro-Dynamic Programming

An overview of pricing models for revenue management

Manufacturing and Service Operations Management

Periodic pricing of seasonal products in retailing

Management Science

Coordinating clearance markdown sales of seasonal products in retail chains

Operations Research

The science of revenue management when passengers purchase the lowest available fare

Journal of Revenue and Pricing Management

Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions

Management Science

Optimal dynamic pricing of inventories with stochastic demand over finite horizons

Management Science

A multi-product, multi-resource pricing problem and its applications to network yield management

Operations Research