A multiple class seat allocation model with replenishment

In this paper we consider an airline seat allocation model where multiple fare classes are available and replenishment can be allowed for the lower fare classes. Such a replenishment likely occurs whenever the higher fare classes do not have demands large enough compared with the number of the pre-allocated seats. Then, the lower fare classes should be reopened while the prices may be discounted during the periods left before the departure. If the demands for high fare classes are large enough, the airlines should not discount the prices and the replenishment may not be allowed from the point of view of revenue management. We also allow the customers to cancel their reservations. Our model formulation is much closer to the airlines practice. We show under some conditions that the expected revenue with replenishment is greater than the one without replenishment and that there exists a simple optimal booking policy. Some numerical examples are provided to confirm analytical properties of the revenue function as well as of the optimal policy.