This paper considers the problem of solving conflicts arising among several aircraft that are assumed to move in a shared airspace. Aircraft can not get closer to each other than a given safety distance in order to avoid possible conflicts between different airplanes. For such system of multiple aircraft, we consider the path planning problem among given waypoints avoiding all possible conflicts. In particular we are interested in optimal paths, i.e. we want to minimize the total flight time. We propose two different formulations of the multi-aircraft conflict avoidance problem as a mixed-integer linear program: in the first case only velocity changes are admissible maneuvers, in the second one only heading angle changes are allowed. Due to the linear formulation of the two problems, solutions may be obtained quickly with standard optimization software, allowing our approach to be implemented in real time.