Technical Note—Revenue Management with Calendar-Aware and Dependent Demands: Asymptotically Tight Fluid Approximations

When modeling the demand in revenue management systems, a natural approach is to focus on a canonical interval of time, such as a week, so that we forecast the demand over each week in the selling horizon. Ideally, we would like to use random variables with general distributions to model the demand over each week. The current demand can give a signal for the future demand, so we also would like to capture the dependence between the demands over different weeks. Prevalent demand models in the literature, which are based on a discrete-time approximation to a Poisson process, are not compatible with these needs. In this paper, we focus on revenue management models that are compatible with a natural approach for forecasting the demand. Building such models through dynamic programming is not difficult. We divide the selling horizon into multiple stages, each stage being a canonical interval of time on the calendar. We have a random number of customer arrivals in each stage whose distribution is arbitrary and depends on the number of arrivals in the previous stage. The question that we seek to answer is the form of the corresponding fluid approximation. We give the correct fluid approximation in the sense that it yields asymptotically optimal policies. The form of our fluid approximation is surprising as its constraints use expected capacity consumption of a resource up to a certain time period conditional on the demand in the stage just before the time period in question. Letting K be the number of stages in the selling horizon, c min be the smallest resource capacity, and ϵ be a lower bound on the mass function of the demand in a stage, we use the fluid approximation to give a policy with a performance guarantee of 1 − O ( ( c min + K / ϵ 6 ) log c min c min ) . As the resource capacities and the number of stages increase with the same rate, the performance guarantee converges to one. To our knowledge, this result gives the first asymptotically optimal policy under dependent demands with arbitrary distributions. When the demands in different stages are independent, letting σ 2 be the variance proxy for the demand in each stage, a similar performance guarantee holds by replacing 1 ϵ 6 with σ 2 . Our computational work indicates that using the right fluid approximation can make a dramatic impact in practice.