Fleet mix problem

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To determine how many of each type of aircraft should be in a fleet to meet demands and minimize total cost. Here is an integer linear program model:

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).


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