A Polynomial-time Algorithm for the Large Scale of Airplane Refueling Problem

Airplane refueling problem is a nonlinear unconstrained optimization problem with $n!$ feasible solutions. Given a fleet of $n$ airplanes with mid-air refueling technique, the question is to find the best refueling policy to make the last remaining airplane travels the farthest. In order to deal with the large scale of airplanes refueling instances, we proposed the definition of sequential feasible solution by employing the refueling properties of data structure. We proved that if an airplanes refueling instance has feasible solutions, it must have the sequential feasible solutions; and the optimal feasible solution must be the optimal sequential feasible solution. Then we proposed the sequential search algorithm which consists of two steps. The first step of the sequential search algorithm aims to seek out all of the sequential feasible solutions. When the input size of $n$ is greater than an index number, we proved that the number of the sequential feasible solutions will change to grow at a polynomial rate. The second step of the sequential search algorithm aims to search for the maximal sequential feasible solution by bubble sorting all of the sequential feasible solutions. Moreover, we built an efficient computability scheme, according to which we could forecast within a polynomial time the computational complexity of the sequential search algorithm that runs on any given airplanes refueling instance. Thus we could provide a computational strategy for decision makers or algorithm users by considering with their available computing resources.