In project scheduling under processing times uncertainty, the Anchor-Robust Project Scheduling Problem is to find a baseline schedule of bounded makespan and a max-weight subset of jobs whose starting times are guaranteed. The problem was proven NP-hard even for budgeted uncertainty. In the present work we design mixed-integer programming (MIP) formulations that are valid for a variety of uncertainty sets encompassing budgeted uncertainty. A new dominance among solutions is proposed, resulting into an MIP formulation. We further study the combinatorial structure of the problem. Non-trivial polynomial cases under budgeted uncertainty are exhibited, where the dominance-based formulation yields a polyhedral characterization of integer solutions. In more general cases, the dominance-based formulation is shown to be tighter than all previously known formulations. In numerical experiments we investigate how the formulation performs on instances around the polynomial cases, for both budgeted uncertainty sets and more elaborate uncertainty sets involving several budgets.

在处理时间不确定的项目调度中，Anchor-Robust 项目调度问题是找到一个有界制造跨度的基线调度和一个最大权重的作业子集，其开始时间是有保证的。即使对于预算不确定性，该问题也被证明是 NP-hard 问题。在目前的工作中，我们设计了混合整数规划 (MIP) 公式，这些公式适用于包括预算不确定性在内的各种不确定性集。提出了解决方案中的新优势，从而形成了 MIP 公式。我们进一步研究了问题的组合结构。展示了预算不确定性下的非平凡多项式情况，其中基于优势的公式产生整数解的多面体特征。在更一般的情况下，显示基于优势的公式比所有以前已知的公式更紧密。在数值实验中，我们研究了公式如何在多项式情况下的实例上执行，包括预算不确定性集和涉及多个预算的更复杂的不确定性集。