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. We conclude by showing 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.

翻译：我们研究了预算不确定性模型下的鲁棒置换流水车间问题，其中每台机器上最多有给定数量的工件加工时间可能发生偏离。我们证明，该问题的解可通过求解多项式数量的对应标称问题实例来确定。作为直接推论，我们的结果表明该鲁棒流水车间问题在两台机器情况下可在多项式时间内求解，对于任意固定数量的机器可在多项式时间内近似求解。我们主要结果的归约推导采用了与Bertsimas和Sim（2003）类似的分析方法，不同之处在于对偶化应用于最小-最大目标函数的各项而非线性目标函数。考虑到文献中已开发出基于启发式和精确整数规划的方法来解决双机流水车间问题，我们的结果可能令人惊讶。最后，我们展示了两台机器和三台机器情况下，通过朴素归约到标称问题所带来的整体运行时间的对数级改进。