On the structural properties of a discrete-time single product revenue management problem

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Abstract

We consider a multi-period revenue management problem in which multiple classes of demand arrive over time for the common inventory. The demand classes are differentiated by their revenues and their arrival distributions. We investigate monotonicity properties of varying problem parameters on the optimal reward and the policy.

Introduction

We consider a finite horizon, single-product inventory control problem in which the decision-maker accepts or rejects customer requests coming from multiple demand classes. Customer requests can be for multiple units of the product (batch orders) and we allow partial fulfillment of demand for accepted requests. The optimal decisions depend on factors such as the available inventory, relative profitability of demand classes, projected volume and mix of future demand (distribution of future demand), and time-to-go till the end of the time horizon. Clearly, this is a typical revenue management problem, which has garnered great interest from researchers (see Talluri and van Ryzin [1] for a comprehensive survey of revenue management literature). Revenue management has also become a very powerful managerial tool to exploit the revenue-enhancement potential in many businesses (Cross [2], Smith et al. [3]). Revenue management empowers these businesses to effectively address the challenges of matching supply and demand. However, due to the many difficulties in successful implementation of revenue management systems, companies have not been able to fully realize the benefits from the cutting-edge tools that researchers have developed in the last few decades (Lahoti [4]). One of these obstacles is the estimation of parameters used in the underlying revenue management models, which govern the principles of how to allocate and reserve resources for high profit customers. Understanding the impact of each parameter on optimal admission policies is a key factor in successful revenue management practices, as it allows managers to perform what-if analysis when faced with changing parameter values (possibly because of estimation errors). This paper focuses on the structure of optimal admission policies in a well-established model of dynamic revenue management. In particular, the investigation focuses on the effects of perturbations of the problem parameters on the optimal admission policy and the optimal reward. The parameters of interest include arrival probabilities of different classes as well as their rewards. Such an investigation is crucial for designing admission policies which are robust to changes in the parameters.

Revenue management literature has gone a long way in establishing the structure of optimal policies and our results complements some of the existing results. Here, we only review work that is particularly related to our paper. The structure of the optimal admission policy for the basic dynamic revenue management is established in Lee and Hersh [5]. Notably, Lee and Hersh establish the optimality of a nested threshold-type admission policy. These results were later streamlined and generalized by Lautenbacher and Stidham [6]. Brumelle and Walczak [7] obtain further results in the challenging semi-Markov case. In the presence of inaccurate estimates of customer arrival distributions, Birbil et al. [8] illustrate, using simulations, the benefit of using robust optimization techniques in reducing expected revenue variability. Talluri and Van Ryzin [1] present a summary of the most important results in both the static and dynamic versions of the problem.

Among the rich literature investigating dynamic policies in revenue management, the work of Lautenbacher and Stidham [6] is of particular importance for us for two reasons. First, we borrow their model which is versatile and subsumes some of the well-established dynamic and static models in the literature. Second, Lautenbacher and Stidham emphasize the significance of focusing on the monotonicity of certain operators appearing in the value function of the dynamic program. This is also the approach we take for investigating the monotonicity properties related to changes in the problem parameters. Koole [9] presents an excellent overview of monotonicity results in Markov Decision Process with applications in queueing. In particular, for varying arrival probabilities, we adapt some of the recent ideas from Çil, Örmeci and Karaesmen [10] to the model of [6]. The main focus of [10] is on continuous-time infinite-horizon queueing-inventory models with stationary parameters. The discrete-time model with non-stationary parameters considered here poses additional challenges but it turns out that corresponding results can be obtained. We also obtain additional results pertaining to parameter effects that are particular to the discrete-time revenue management setting. To our knowledge, the only other paper that investigates related monotonicity issues in a revenue management context is Cooper and Gupta [11]. That paper investigates the effect of demand distributions on the expected optimal reward in a single-period setting.

Designing robust policies for revenue management when problem parameters are uncertain has recently attracted attention. For instance, Lan, Gao, Ball and Karaesmen [12] consider the case with limited demand information employing ideas from competitive analysis of on-line algorithms. We do not explicitly consider the challenging robust policy design problem in this paper but our results provide basic guidelines as to what sort of changes in the optimal policy are anticipated as problem parameters are varied within a given uncertainty set.

Section snippets

Model

The model that we employ was first introduced, to our knowledge, by Lautenbacher and Stidham [6]. Suppose that time is divided into decision periods such that at most one request is received in any given period but the customer can demand more than one unit of the product. Let K be the number of decision periods. Time is indexed by k in our model, where k=K is the first period and k=1 is the last period after which all inventories perish. There are n demand classes, with Class i offering to pay

Structural properties

In this section, we first describe a number of basic structural properties of the model in Section 2 that are well known in revenue management literature. We then present our results on the impact of varying two particular problem parameters–the arrival probabilities and the rewards. In this section, we also use numerical examples to illustrate interesting policy implications of our analytical results.

Extensions and discussion

Certain extensions to the model are straightforward. For instance, all of the properties go through if non-zero salvage values are assumed at the end of the horizon as long as the salvage value function satisfies the required properties for induction (i.e. concavity in the inventory level). Holding costs can also be handled in a straightforward manner for most properties as long as the holding cost function is increasing and convex. On the other hand, since the holding cost function applies

Acknowledgements

This research was partially supported by TUBITAK and the TUBA-GEBIP programme. F. Karaesmen is grateful to the Dept. of Ind. Eng. and Man. Sci. of Northwestern University where part of this research was done.

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