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.

翻译：飞机空中加油问题是一个具有$n!$个可行解的非线性无约束优化问题。给定一个配备空中加油技术的$n$架飞机机队，问题在于寻找最佳的加油策略，使得最后一架剩余飞机飞行距离最远。为处理大规模飞机加油实例，我们通过利用数据结构中的加油特性，提出了序列可行解的定义。我们证明了若某个飞机加油实例存在可行解，则必存在序列可行解；且最优可行解必为最优序列可行解。进而提出由两步组成的序列搜索算法：第一步旨在找出所有序列可行解，当输入规模$n$大于某个索引数时，我们证明了序列可行解的数量将以多项式速率增长；第二步通过冒泡排序对所有序列可行解进行排序，以搜索最大序列可行解。此外，我们构建了一个高效可计算性方案，据此可在多项式时间内预测针对任意给定飞机加油实例运行的序列搜索算法的计算复杂度。由此，我们可根据决策者或算法用户拥有的计算资源，为其提供计算策略。