Abstract This paper studies a robust (min–max regret) single machine scheduling problem with uncertain processing times represented by interval data. The objective is to obtain robust sequences of jobs that minimize the absolute deviation of total tardiness from the optimal solution under the worst-case scenario. The problem is first formulated as a mixed-integer linear programming (MILP) model and assuming that the corresponding deterministic NP-hard problem for the mid-point scenario can be solved optimally, an optimal schedule under the mid-point scenario is then proved to be a 2-approximation solution to the problem. Next, the worst-case scenarios are proved to be not necessarily at the upper or lower limits of the interval processing times. Utilizing these results, a 2-approximation algorithm (2AA), a worst-case scenario-based heuristic (WSH), and an approximate worst-case-based heuristic (AWH) are proposed and empirically evaluated, in which an iterative procedure for evaluating the maximum regret of a solution is integrated. Finally, the paper is concluded by suggesting some fruitful directions for future research in this area.

中文翻译：

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具有间隔处理时间和总延迟标准的稳健（最小-最大遗憾）单机调度

摘要 本文研究了具有不确定处理时间的鲁棒（最小-最大遗憾）单机调度问题，该问题由区间数据表示。目标是获得健壮的作业序列，在最坏的情况下最大限度地减少总延迟与最佳解决方案的绝对偏差。该问题首先被表述为混合整数线性规划 (MILP) 模型，并假设中点场景的相应确定性 NP-hard 问题可以最优求解，然后证明中点场景下的最优调度为是问题的 2-近似解。接下来，证明最坏情况不一定在间隔处理时间的上限或下限。利用这些结果，一个 2-近似算法 (2AA)，一个基于最坏情况场景的启发式算法 (WSH)，并提出了一种近似的基于最坏情况的启发式（AWH）并进行了经验评估，其中集成了用于评估解决方案最大遗憾的迭代过程。最后，本文最后提出了该领域未来研究的一些富有成果的方向。