Solution to the continuous time dynamic yield management model

We formulate the yield management problem as a continuous time, stochastic, dynamic programming model. We derive an expression for the expected revenue in terms of the stochastic booking processes and the control policies. The solution to the problem is found by maximizing the expected revenue over the possible control decisions. The solution is for an, arbitrary number of fare classes and arbitrary booking curves. In particular, it requires no assumptions on the order of arrivals from different fare classes. The solution can be expressed in terms of a double recursion complex. At each node of the complex, the upper limit of a one-dimensional integral is solved to find a critical time for each fare class and for each value of remaining capacity. The critical times are the only values that need to be stored in the reservation control system to achieve optimal real-time control. This simple result is somewhat surprising given the complexity of even the static programming versions of the problem. We derive simple expressions of expected revenues and bid prices, which provide useful information to the user of a yield management system.