Optimal solution for a cargo revenue management problem with allotment and spot arrivals

We consider a single-leg cargo revenue management problem, in which a two-dimensional cargo capacity is sold through allotment contracts and in the spot market. Capacity sold on an allotment basis is guaranteed. We optimally solve the problem of determining how much of the total weight and volume capacity to sell on an allotment basis, by deriving a closed-form expression of the objective function. We provide numerical examples of industry-size problems and perform sensitivity analysis by changing some problem parameters. The sensitivity analysis illustrates the dependency of the optimal decisions on the spot and allotment booking types. The remaining capacity is then sold over a booking horizon in the spot market. Allotment bookings and spot requests can arrive any time over the booking horizon. Since some of the allotment bookings might not show up at departure, cargo carriers tend to overbook the remaining capacity allocated to spot requests. For these requests, we formulate a discrete-time dynamic capacity control model, to decide which of the spot requests to accept, based on the total weight and volume of the allotment show-ups and spot bookings accepted at the time of an arrival. We solve the exact dynamic programming model for medium-size industry problems. Since the booking policy based on critical booking levels or time periods is not optimal, we propose several heuristics to solve large industry problems and derive an upper bound on the value function. We test their performance via simulation against the optimal solution, the upper bound, and the first-come first-served policy, and recommend a heuristic that performs well in a wide variety of numerical cases. Finally, we show via simulation, that our model outperforms the one existing in the literature, for small and medium-size industry problems.