A polynomial-time algorithm to solve the large scale of airplane refueling problem

Airplane refueling problem (ARP) is a scheduling problem with an objective function of fractional form. Given a fleet of $n$ airplanes with mid-air refueling technique, each airplane has a specific fuel capacity and fuel consumption rate. The fleet starts to fly together to a same target and during the trip each airplane could instantaneously refuel to other airplanes and then be dropped out. The question is how to find the best refueling policy to make the last remaining airplane travels the farthest. We give a definition of the sequential feasible solution and construct a sequential search algorithm, whose computational complexity depends on the number of sequential feasible solutions referred to $Q_n$. By utilizing combination and recurrence ideas, we prove that the the upper bound of $Q_n$ is $2^{n-2}$. Then we focus on the worst-case and investigate the complexity of the sequential search algorithm from a dynamic perspective. Given a worst-case instance under some assumptions, we prove that there must exist an index $m$ such that when $n$ is greater than $2m$, $Q_n$ turns out to be upper bounded by $\frac{m^2}{n}C_n^m$. Here the index $m$ is a constant and could be regarded as an "inflection point": with the increasing scale of input $n$, $Q_n$ turns out to be a polynomial function of $n$. Hence, the sequential search algorithm turns out to run in polynomial time of $n$. Moreover, we build an efficient computability scheme by which we shall predict the complexity of $Q_n$ to choose a proper algorithm considering the available running time for decision makers or users.