Revenue management with flexible products: The value of flexibility and its incorporation into DLP-based approaches

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Abstract

A major benefit of flexible products is that they allow for supply-side substitution even after they have been sold. This helps improve capacity utilization and increase the overall revenue in a stochastic environment. As several authors have shown, flexible products can be incorporated into the well-known deterministic linear program (DLP) of revenue management׳s capacity control. In this paper, we show that flexible products have an additional “value of flexibility” due to their supply-side substitution possibilities, which can be captured monetarily. However, the DLP-based approaches proposed so far fail to capture this value and, thus, steadily undervalue flexible products, resulting in lower overall revenues. To take the full potential of flexible products into account, we propose a new approach that systematically increases the revenues of flexible products when solving the DLP and performing capacity control. A mathematical function of variables available during the booking horizon represents this artificial markup and adapts dynamically to the current situation. We determine the function׳s parameters using a standard simulation-based optimization method. Numerical experiments show that the benefits of the new approach are biggest when low value demand arrives early. Revenues are improved by up to 5% in many settings.

Introduction

In this paper, we reconsider the well-known revenue management problem of optimal capacity control with flexible products. Flexible products allow the provider to decide on the utilized resources sometime after selling the product. Gallego and Phillips (2004) were the first to introduce the problem to the academic literature. They were motivated by the popularity of these supply-side substitution opportunities and the high practical relevance resulting from them, for example, in the context of airlines, hotels, and cruise lines. Moreover, Gallego et al. (2004) incorporated flexible products into the common dynamic programming approach for network revenue management. Similar to the standard setting, the dynamic program is computationally intractable even for modest problem sizes due to the multidimensional state space that must be considered. Researchers have therefore investigated different types of approximations in a considerable number of follow-up papers. The most prominent approximations employ a deterministic linear program (DLP), which is quite common in practical applications. It is obtained by simply replacing any uncertainty in the dynamic program with expected values.

As we demonstrate in this paper, even though many researchers have followed DLP-based approaches, the straightforward extension of the DLP does not take the full potential of flexible products into consideration and has additional drawbacks compared to its application in the standard setting. More precisely, our contributions are as follows: First, given the dynamic program of Gallego et al. (2004), we analytically isolate the additional monetary “value of flexibility” that comes along with acceptance of flexible requests. Second, we show that none of the DLP-based approaches proposed so far considers this value. Therefore, flexible products׳ benefits are systematically underestimated and the resulting control mechanisms are too restrictive regarding the acceptance of flexible requests. Third, we propose a new and straightforward DLP-based approach that avoids the strict preference of regular products and, by using simulation-based optimization, better incorporates the benefits of flexible products. An extensive numerical study demonstrates the applicability of this approach and shows that, in most settings, it significantly outperforms existing approaches in terms of the overall achievable revenue.

The remainder of this paper is structured as follows: In Section 2, we accurately restate the revenue management problem of optimal capacity control with flexible products, including the assumptions made. Furthermore, we carve out the relevance of the problem by providing examples from various industries and review the relevant scientific literature. In Section 3, we summarize the standard models for network revenue management with flexible products. Based on this, we begin Section 4 with the analytical derivation and investigation of the value of flexibility in the dynamic program. We then turn to the DLP-based approximations and show why they have additional shortcomings with regard to flexible products. In Section 5, we present our new, improved approach and investigate its performance computationally in Section 6. In Section 7, we discuss the potential limitations of the chosen approach, as well as the assumptions, and conclude the paper.

Section snippets

Problem statement, practical relevance, and related literature

In this section, we provide an overview of revenue management with flexible products. We first state the problem of capacity control with flexible products in detail. Using various examples from different industries, we then show the problem׳s relevance in practice. Finally, we extensively review the relevant scientific literature.

Basic model formulations

In this section, we briefly summarize the standard models for network revenue management with flexible products from the literature. First, we introduce the relevant notation. We then restate the standard DP for revenue management with flexible products and specify the optimal control policy. Finally, we state the corresponding DLP approximation.

Value of flexibility

In this section, we show that flexible products possess an additional value that can be captured monetarily. We call this their “value of flexibility,” which is analytically derived from the corresponding DP formulation and further investigated. We then turn to the DLP and show that this model completely fails to capture the value of flexibility because it ignores the possibility of substitution after the sale. This leads to a severe and systematic shortcoming of existing DLP-based approaches

Simulation-based optimization approach for controls based on the DLP approximation

To tackle the limitations of DLP-based approaches discussed in the previous section, we propose a generic approach that aims at incorporating an approximation of the value of flexibility into the DLP, using simulation-based optimization. Furthermore, we show how the new approach can be applied to bid price controls. Again, Table 3 provides an overview of the notation introduced in this section.

Numerical experiments

In this section, we present the results of an extensive simulation study. The simulations were conducted on an Intel Xeon processor-based PC (Intel Core i7-2600 CPU, 4 Cores, 3.40 GHz, 8 GB RAM, Microsoft Windows 7 Enterprise SP1). All the algorithms were implemented in Matlab (Version 8, Release R2012b), using the Optimization Toolbox (in particular, the integrated linear programming routine linprog) and the Global Optimization Toolbox (in particular, the simulation-based optimization routine

Conclusion

Having analyzed the numerical results in detail in the previous section, we now first take a broader perspective and discuss the assumptions made, as well as the methodology used. Finally, we summarize the main results and point out aspects worthy of future research.

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