In addition to the omnipresent irradiation from galactic cosmic rays (GCR) and their secondary products, passengers and aircraft crew may be exposed to radiation from solar cosmic rays during ground level enhancements (GLE). In general, lowering the flight altitude and changing the flight route to lower latitudes are procedures applicable to immediately reduce the radiation exposure at aviation altitudes. In practice, however, taking such action necessarily leads to modifications in the flight plan and the consequential, additional fuel consumption constrains the mitigating measures. In this work we investigate in a case study of the ground level event of December 13th 2006 how potential mitigation procedures affect the total radiation exposure during a transatlantic flight from Seattle to Cologne taking into account constraints concerning fuel consumption and range.