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A risk-sensitive model for managing perishable products.

Article Excerpt
This article presents a risk-sensitive model for managing perishable products assuming the supplier is averse to the variation of revenues. While traditional risk-neutral revenue management models offer optimal strategies in the long, run, they are exposed to the variation of revenue flows. If a short-term revenue target is a primary concern for the supplier, the risk-neutral assumption fails to provide the best policy needed. The proposed model uses an exponential function with a risk-sensitive parameter instead of the conventional risk-neutral objective. The risk parameter measures how the supplier is sensitive to the deviation of revenues. We show that the new objective function captures the supplier's risk behavior. We develop a recursive procedure for the optimal solution in closed form. The optimal policy has attractive properties such as nested active price set, monotonicity with respect to the remaining time and inventory, and threshold-type control. When the supplier is more sensitive to the uncertain revenue flows, the risk-sensitive model leads to more conservative pricing policies. Finally, we show that the risk-neutral model is a special case of the proposed framework.

Subject classifications: revenue management; risk sensitive; exponential utility.

Area of review: Manufacturing, Service, and Supply Chain Operations.

History: Received May 2004; revisions received December 2004, October 2005, November 2006, March 2007; accepted June 2007.

1. Introduction

Traditional revenue management develops optimal pricing/inventory control policies based on the assumption that decision makers are risk neutral. Hence, the objective of these models is to maximize the expected revenues at the end of the sales period. The risk-neutral assumption works well when the strategy is implemented in the long run. Oftentimes management is concerned about annual or quarterly revenue performance which involves a large number or runs under similar conditions. However, because of the random nature of demand, forecast errors, and unexpected capacity changes, the revenue realized at the end of each period varies and may significantly differ from its expected value. There are various compelling reasons that short-term revenue is a main consideration for managers--for example, a requirement of meeting certain financial constraints, a good chance of achieving revenue goals during specific time windows, shareholder expectations of short-term performances, etc. A poor performance in a single run can be critical to the short-term goal and may not be compromised by the long-term optimality. Hence, these risk-neutral models may not be optimal for companies in the short run. It is not uncommon that managers choose more conservative pricing/inventory policies to ensure a higher probability of attaining the revenue target. To reflect this business practice, a model that incorporates management's risk behavior is needed. The objective of such a model will not simply maximize the expected revenue, as employed in the risk-neutral models.

Research on risk behavior in revenue management is sporadic. As Bitran and Caldentey (2003, p. 224) note: "essentially all the (RM) models that we have discussed assume that the seller is risk neutral." Recently, integration of risk and revenue management has gained increasing attention among researchers. Lim and Shanthikumar (2007) consider uncertainty resulting from forecast errors in revenue management. By relating uncertainty to stochastic differential games, they derive a dynamic pricing policy and its structural properties. (1) Levin et al. (2007) propose a model for pricing perishable services or products that incorporates risk behavior in revenue management. To control the probability that total revenue falls below a minimum acceptable level, they introduce a risk factor into the model and derive structural properties of the optimal policies.

In this article, we develop a risk-sensitive model to address the concern of uncertain cash streams in revenue management. Risk-sensitive control has been studied in optimal investment models (Fleming 1995, Fleming and Sheu 2000) and nonlinear stochastic filtering problems such as hidden Markov model estimation (Dey and Moore 1997, Remezani and Marcus 2005), among others. Following a widely adopted approach in the literature, we replace the conventional risk-neutral objective by an exponential utility function. There are several reasons for choosing the exponential utility function. First, as shown in [section]2, it captures the essence of uncertainty and risk. When the function is expanded by a Taylor series, the linear portion approximates the expected revenue, while the higher-order moments describe the risk. Second, the exponential utility function is tractable and enables us to derive the optimal solution in closed form and its structural properties. Third, it is easy to derive the equivalent expected revenue from the utility function. Hence, the alternative objective function offers a nice connection between theoretical analysis and practical needs.

We show that the...