The hotel yield management with two types of room prices, overbooking and cancellations

We consider a problem for hotel room allocations with early discount, cancellations and overbooking. In our model, customers can book a room in discounted price if they reserve before a certain deadline. The model takes overbooking into account. We have expressed the expected total sale function, proved it is unimodal on the number of allocated rooms for early discount and for overbooking respectively, under a condition and clarified the range where the optimal solution exists. We have derived an optimal allocation for early discount in a specified model without cancellations and overbooking.