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Providing Internet service above the clouds is of ever-increasing interest and in this context aeronautical {\it{ad-hoc}} networking (AANET) constitutes a promising solution. However, the optimization of packet routing in large ad hoc networks is quite challenging. In this paper, we develop a discrete $ε$ multi-objective genetic algorithm ($ε$-DMOGA) for jointly optimizing the end-to-end latency, the end-to-end spectral efficiency (SE), and the path expiration time (PET) that specifies how long the routing path can be relied on without re-optimizing the path. More specifically, a distance-based adaptive coding and modulation (ACM) scheme specifically designed for aeronautical communications is exploited for quantifying each link's achievable SE. Furthermore, the queueing delay at each node is also incorporated into the multiple-objective optimization metric. Our $ε$-DMOGA assisted multiple-objective routing optimization is validated by real historical flight data collected over the Australian airspace on two selected representative dates.