Maximizing the average fuel savings for aircraft fleet under a fixed periodic schedule

We consider the problem of assigning a fleet of aircraft of different types to a given periodic flight schedule aiming to maximize fuel economy for a predetermined fleet flight schedule. The challenge is to select the most profitable aircraft type for each flight and then determine the required number of aircraft and the number of deadhead flights to maximize fuel economy. We split this general problem into two successive optimization problems. The first one, the aircraft-type assignment problem, is to select the most profitable aircraft type for each flight; the second one is the aircraft routing problem. One way to solve the first problem, which we adopt in this paper, is to reduce it to a parametric minimum-cost network flow problem. To solve it, we propose a new Newton-type algorithm that finds good solutions for large-scale problems. Another focus of the paper is on efficient solution of the aircraft routing problem for a single aircraft type. The goal is to maximize fuel economy, taking into account both the required number of aircraft and the number of deadhead flights. We transform it into a special fractional assignment problem and solve using a new version of the Newton-type algorithm that is adapted for the aircraft fleet routing problem and runs in O(n⁴) time, where n is the number of flights.