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In this paper, we deal with the problem of a fixed number of units of a certain perishable commodity over a continuous time horizon. Airline seats, hotel rooms, advertising space in newspapers, and some seasonal products that must be sold before the end of the season are examples of such commodities that cannot be carried over and are not storable for consumers. This paper considers such a problem of continuous time perishable inventory control by applying semi-Markov decision processes over a finite time horizon. It is shown that there is an optimal policy that is simple and stationary. Furthermore, some analytical properties of this optimal policy and its value function are explored under certain specific assumptions.

Consider a certain perishable commodity whose supply is fixed. A company wants to sell a fixed number of units of this commodity. On the other hand, consumers have different (nonhomc- geneous) preferences for the commodity. In such a circumstance, the company can set different prices on the identical commodity; that is, the law of one price does not hold in this market. Such commodities are called price differentiation products. Examples of price differentiation products might include airline seats, hotel rooms, concert tickets, cabins on cruise ships, and seasonal goods such as winter coats that have to be sold before the end of the season. The date and place for selling such commodities is fixed, it is not possible to use these commodities either before or after the specified date, and they are perishable goods that cannot be carried over. It is possible, however, to sell these commodities beforehand. In this paper, we develop a framework of inventory control for perishable commodities with limited supply by explicitly incorporating the existence of multiple prices for that commodity. It is shown that such an inventory control model can be formulated as a semi-Markov decision process over a finite continuous time horizon. Our main result is to show that, under some specific assumptions, there is an optimal policy and an optimal value function corresponding to it. In addition, some analytical properties are explored. The problem of selling a fixed number of units of a certain perishable commodity in order to maximize the expected revenue is known as the yield management problem in the airline industry (see [i-3]), w h ere most studies on yield management have focused on either pricing policy or static control problems with two fares (see [1,4-71). Feng and Gallego [S] consider the problem of deciding the optimal timing of a single-price change from a given initial price to either a lower or higher price. Gallego and Ryzin [5] investigate the problem of dynamic pricing policies for selling a given stock of items by a deadline. Lee and Hersh [9] develop a discrete-time dynamic programming model for finding an optimal booking policy for airline seat inventory control. Brumelle and Walczak [lo] model an airline seat allocation for a single-leg flight with multiple fares that is similar to ours but has different assumptions. The present study differs from these earlier studies in two regards. First, the planning horizon is continuous times; second, ours is a dynamic model that deals with a multiple-price selling policy for a perishable product that is not necessarily restricted to airline seats. In Section 2, we develop the problem of selling a fixed number of units of a perishable commod- ity over a continuous time planning horizon. We apply a semi-Markov decision process approach to the problem so as to maximize the expected total revenue. Section 3 discusses what analytical properties an optimal policy and its value function possess. In particular, we are interested in the development of value function concavity and optimal policy monotonicity with respect to their arguments; these tell us that an optimal selling curve decreases as the time draws nearer to the end of the planning horizon. Finally, the last section is a conclusion with additional comments.

In this paper, we have extended the familiar single-period seat inventory control model into a dynamic model over continuous time and have carried out its formularization within the frame- work of a semi-Markov decision process. Our finding is that an optimal policy and its expected revenue exist in regard to a demand type that arrives in accord with a semi-Markov process, and we have shown the analytical properties of that property. We have shown in particular (in Theorem 2) that, under the condition that the transition probability does not depend on the price of the demand type, the demand size of a higher price is greater than that of a lower price. In other words, the reservation limit has a nested structure with regard to price. Being able to prove that an optimal policy exists within a demand policy class that has a simple structure is surely extremely convenient from a business standpoint. It means, for example, that an accepted limit curve I* (I, C#J, t) of the control limit model exists, and if 1 5 1* (1,4, t) one stops demand acceptance, whereas if I > I*(I, 4, t), one sells min{s, I - I*(I, 4, t)} units of that commodity. Furthermore, it is realistically desirable that this 1*(1,4, t) is a nonincreasing function of t. Models made up of several (three or more) price classes that explicitly take into account upgrading and overbooking remain for future studies. If we limit the arrival process of consumers and the classes of demand acceptance policies, it might be possible to deduce the above properties, but as soon as one deals with three or more prices the problem of analysis rapidly becomes difficult. Th ere are several directions in which future research in this area could be conducted. One would be to attempt a combination of an estimate of the customer demand function and an inventory allocation policy; this is an important real problem for a firm or company that cannot directly observe the demand function. Likewise, the problem of differentiating commodities with different prices in the most desirable form by imposing restraints on their use at every price level is another important problem from the point of view of marketing.