Capacity Control in a Random Environment

Abstract

We consider a non-stop flight of an airplane with a capacity of C that is to depart after a certain time T. There are i_max (i_max ∈ ℕ) booking classes, i = 1, . . ., i_max, with associated fares ϱ_i ordered such that $0 < \rho _{i_{max}} < \rho _{i _{max}} - 1 < \ldots < \rho _i$ . The number of booking periods in [0; T] is given by some external process and might be random. In every booking period n, a customer requests a certain number of reservations $d_n \in \mathfrak{D}: = \{ 0,1,...,d_{max} \} ,d_{max} \in \mathbb{N}_0$ , for seats of booking class $i_n \in \mathfrak{J}: = \left\{ {0,1,...,i_{max} } \right\}$ . (i_n = 0 with ϱ₀ = 0 denotes an artificial booking class corresponding to no customer request.) Thus, the (n + 1)st customer request provides information on the number $d_n \in \mathfrak{D}$ of reservations (the customer is interested in) and the booking class $i_n \in \mathfrak{J}$ with reward $\rho _i {_n }$ that is offered for each of the d_n reservations. It must be decided how many of these requested reservations should be actually sold.