Adaptive Capacity Allocation with Censored Demand Data: Application of Concave Umbrella Functions

One of the classical problems in revenue management is the capacity allocation problem, where the manager must allocate a fixed capacity among several demand classes that arrive sequentially in the order of increasing fares. The objective is to maximize expected revenue. For this classical problem, it has been known that one can compute the optimal protection levels in terms of the fares and the demand distributions. Contrary to conventional approaches in the literature, we consider the capacity allocation problem when the demand distributions are unknown and we only have access to historical sales, which represent censored demand data. We develop an adaptive algorithm for setting protection levels based on historical sales, show that the average expected revenue of our algorithm converges to the optimal revenue, and establish the rate of convergence. Our algorithm converges faster than any previously known algorithm for this problem. Our analysis relies on a novel concept of a concave umbrella function, which provides a lower bound for the revenue function while achieves the same maximizer and the same maximum value. Extensive numerical results show that our adaptive algorithm performs well. ∗Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. E-mail: [email protected]. †School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. E-mail: [email protected]