Pricing and Dynamic Capacity Allocation

Two classes of customers

• The price charged to the economy customers is $p_e$ and that for business travellers is $p_b$. Of course, $0 < p_e < p_b$.
• Hotel opens the room booking for economy class first and then in the last phase of booking (possibly when the economy class customers have been exhausted), the rooms are sold to the business customers.
• The capacity allocation problem: How many rooms should be sold at the economy price $p_e$? Alternatively, the hotel should protect how many rooms for the business customers?
• The tradeoff in the context of revenue and probabilities.

Capacity Allocation Problem

• In the theory of yield management (revenue management), this problem is about finding ‘discount booking limit’ (maximum number of low-priced booking allowed)
• Alternatively, the problem is to find ‘protection level’ for the business bookings. If $C$ is the total capacity of the hotel (# of rooms), and $k$ is the booking limit, then the protection level, $y = C -k$.

(Here, we assume that the room are reserved for the full day. Hence, on any given day, the rooms are available or not. Also, we neglect cancellations and no-shows.)

Tradeoff explained

• Setting booking limit too low - possibility of more economy customers. With no rooms reserved at low price, they may not book. If enough business customers don’t show up last minute, then we have empty rooms at hand. This is called spoilage.
• Setting booking limit too high - We sell a lot of rooms at low price. Some of these customers could actually be business customers with higher WTP. Potential loss of revenue. This is dilution (of revenue).

Basic Algorithm

• Let $F_b(x)$ be the cumulative distribution function of the business category demand. Similarly, for the economy category $F_e(x)$.
• Therefore, let the probability that the demand in business category is more than $x$ is $\overline{F_b}(x) = 1- F_b(x)$. Similarly, $\overline{F_e}(x) = 1- F_e(x)$.

Marginal Revenue

• Start with a minimum vale of the booking limit $k$.
• The marginal change in revenue by adding one more seat to $k$ (that is, $k+1$)