# Outsmarting Human Design in Airline Revenue Management
## 3 Background
### The Single-Leg Problem  
We consider the monopolistic, capacity-constrained single-leg problem. In this problem, there is only one airline operating a single leg from point A to point B. Every day a new flight is open for sale and another flight departs, closing for new bookings. Each flight is open for sale T days prior to departure (also called the booking horizon), and the airline must manage T _active_ flights _simultaneously_. The customers are only willing to book the flight departing on a specific day, and they either choose to book or not to book at all. The RMS's goal, then, is to select the prices for _every_ active flight so as to maximize the airline's long-term revenue. For simplicity, we assume that overbooking and cancellations are not possible and that the flight has a single cabin class (e.g., first class, business, or economy), with C being the flight's capacity for this unique cabin. We consider that the RMS must select the prices from a discrete fare structure \(\mathrm{F}=\{f_{0},\ldots,f_{n-1}\}\), where \(f_{i}\) is a price point. Furthermore, for the sake of generality, from now on we call a \"day\" a \"time step\" (because in principle it could be any unit of time). We assume that demand behaves according to a parametric demand function \(d(f;\psi)\) that returns the expected number of customers willing to pay for fare \(f\), and \(\psi\) represents a generic set of parameters of the demand function. A demand model extensively used in the literature is the exponential model [14; 42], with \(\psi=(\
u,\phi)\), given by  
\[d(f;\
u,\phi)=\
u\cdot\Pr\{\mathrm{purchase}|f\}=\
u e^{-\phi(\frac{f}{f_{0}}- 1)},\]  
where customers arrive according to a Poisson distribution with mean \(\
u\), called the arrival rate parameter, and \(\phi\) is the price sensitivity parameter. The constant \(f_{0}\) is the lowest fare in the fare structure such that the customer's purchase probability is one \(\Pr\{\mathrm{purchase}|f_{0}\}=1\). Perhaps the most straightforward way to simulate this demand model consists of generating customer arrivals for each time step according to the Poisson distribution, and, for each arriving customer, we sample a random value between zero and one, and, if the sampled value is smaller than the purchase probability at the selected fare \(f\), the customer accepts the offer, proceeding with the purchase, otherwise the customer rejects the offer, and no booking is made. As the true demand behavior \((\
u_{*},\phi_{*})\) is unknown by the RMS, it must estimate the demand model parameters from historical booking data, which can be performed with an ordinary least squares or a maximum likelihood estimation [33; 43]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Study** & **Method** & **Demand Model** & **Capacity Constraining** & **Multi-Flight** \\ \hline \multirow{3}{*}{CEP [27]} & \multirow{3}{*}{heuristic} & Bayesian, & \multirow{3}{*}{\(\checkmark\)} & \multirow{3}{*}{\(\checkmark\)} \\  & & non-parametric, & & \\  & & parametric & & \\ \cline{1-1} \cline{3-  
Next, the RMS uses the calibrated demand model \(d(f;\
u,\phi)\) to optimize the pricing policy \(\pi\) for each active flight _independently_ of the others. This pricing policy adjusts the price according to flight's current inventory state (e.g., the remaining capacity and the time steps to departure), that is, when the flight has excess capacity, the policy may decrease prices aiming to capture more customers, on the contrary, when the flight has a deficit of capacity, the policy may increase prices aiming to capture only customers that are willing to book at higher prices.