# Fleet mix problem

$LaTeX: \min c^T x : a \ge Ax \ge b, \; x_j \in \{0,1,...,N_j\},$
where $LaTeX: N_j$ is the number of aircraft of type $LaTeX: j$ available; $LaTeX: A_{ij}$ is the capacity of aircraft type $LaTeX: j$ for mission $LaTeX: i$; $LaTeX: a_i$ is the least number of missions of type $LaTeX: i$ that must be flown; $LaTeX: b_i$ is the greatest number of missions of type $LaTeX: i$ that must be flown. The variables are $LaTeX: x_j$ are the number of aircraft of type $LaTeX: j$ in the fleet, and $LaTeX: c_j$ is its maintenance cost. If the aircraft must be purchased, binary variables are introduced, as $LaTeX: x_j - N_j y_j \le 0$, with a fixed charge, $LaTeX: fy$, in the objective $LaTeX: (f > 0)$. There could be additional constraints, such as a budget on total purchases $LaTeX: (fy \le f_0)$ or on total maintenance $LaTeX: (gx \le g_0)$.