Simulation-Based Optimization of Virtual Nesting Controls for Network Revenue Management

Virtual nesting is a popular capacity control strategy in network revenue management. In virtual nesting, products (itinerary-fare-class combinations) are mapped (“indexed”) into a relatively small number of “virtual classes” on each resource (flight leg) of the network. Nested protection levels are then used to control the availability of these virtual classes; specifically, a product request is accepted if and only if its corresponding virtual class is available on each resource required. Bertsimas and de Boer proposed an innovative simulation-based optimization method for computing protection levels in a virtual nesting control scheme [Bertsimas, D., S. de Boer. 2005. Simulation-based booking-limits for airline revenue management. Oper. Res.53 90--106]. In contrast to traditional heuristic methods, this simulation approach captures the true network revenues generated by virtual nesting controls. However, because it is based on a discrete model of capacity and demand, the method has both computational and theoretical limitations. In particular, it uses first-difference estimates, which are computationally complex to calculate exactly. These gradient estimates are then used in a steepest-ascent-type algorithm, which, for discrete problems, has no guarantee of convergence. In this paper, we analyze a continuous model of the problem that retains most of the desirable features of the Bertsimas-de Boer method, yet avoids many of its pitfalls. Because our model is continuous, we are able to compute gradients exactly using a simple and efficient recursion. Indeed, our gradient estimates are often an order of magnitude faster to compute than first-difference estimates, which is an important practical feature given that simulation-based optimization is computationally intensive. In addition, because our model results in a smooth optimization problem, we are able to prove that stochastic gradient methods are at least locally convergent. On several test problems using realistic networks, the method is fast and produces significant performance improvements relative to the protection levels produced by heuristic virtual nesting schemes. These results suggest it has good practical potential.