We consider the robust permutation flowshop problem under the budgeted uncertainty model, where at most a given number of job processing times may deviate on each machine. We show that solutions for this problem can be determined by solving polynomially many instances of the corresponding nominal problem. As a direct consequence, our result implies that this robust flowshop problem can be solved in polynomial time for two machines, and can be approximated in polynomial time for any fixed number of machines. The reduction that is our main result follows from an analysis similar to Bertsimas and Sim (2003) except that dualization is applied to the terms of a min-max objective rather than to a linear objective function. Our result may be surprising considering that heuristic and exact integer programming based methods have been developed in the literature for solving the two-machine flowshop problem. Next, we show a logarithmic factor improvement in the overall running time implied by a naive reduction to nominal problems in the case of two machines and three machines. We conclude by noting that our reduction appears to have more general consequences for robust optimization problems under budgeted uncertainty having a similar form.

翻译：我们考虑在预算不确定模型下的鲁棒置换流水车间调度问题，其中每台机器上至多有给定数量的工件加工时间可能发生偏差。我们证明，该问题的解可以通过求解多项式个对应的名义问题实例来确定。作为直接推论，该结果表明对于两台机器的情况，该鲁棒流水车间问题可以在多项式时间内求解，而对于任意固定数量的机器则可在多项式时间内近似求解。作为主要结果的归约方法类似于Bertsimas和Sim（2003）的分析，区别在于对偶化被应用于极小极大目标函数的各项而非线性目标函数。考虑到现有文献已针对双机流水车间问题开发了基于启发式和精确整数规划的方法，我们的结果可能令人意外。随后，我们展示了对双机和三机情形下，通过简单归约到名义问题所隐含的整体运行时间可实现对数因子的改进。最后指出，我们的归约方法对具有类似形式的预算不确定鲁棒优化问题可能具有更普遍的意义。