We study ordinal makespan scheduling on small numbers of identical machines, with respect to two parallel solutions. In ordinal scheduling, it is known that jobs are sorted by non-increasing sizes, but the specific sizes are not known in advance. For problems with two parallel solutions, it is required to design two solutions, and the performance of an algorithm is tested for each input using the best solution of the two. We find tight results for makespan minimization on two and three machines, and algorithms that have strictly better competitive ratios than the best possible algorithm with a single solution also for four and five machines. To prove upper bounds, we use a new approach of considering pairs of machines from the two solutions.

我们研究了少量相同机器上的顺序完工调度，涉及两个并行解决方案。在序数调度中，已知作业是按非递增大小排序的，但具体大小是事先未知的。对于具有两个并行解决方案的问题，需要设计两个解决方案，并使用两个解决方案中的最佳解决方案对每个输入测试算法的性能。我们发现两台和三台机器上的完工时间最小化的严格结果，以及对于四台和五台机器而言，与具有单一解决方案的最佳可能算法相比，算法具有严格更好的竞争比。为了证明上限，我们使用一种新方法来考虑两个解决方案中的机器对。