In this paper, we consider an optimization problem involving crashing an activity network under a single disruption. A disruption is an event whose magnitude and timing are random. When a disruption occurs, the duration of an activity that has yet to start—or alternatively, yet to complete—can change. We formulate a two-stage stochastic mixed-integer program, in which the timing of the stage is random. In our model, the recourse problem is a mixed-integer program. We prove the problem is NP-hard, and using simple examples, we illustrate properties that differ from the problem’s deterministic counterpart. Obtaining a reasonably tight optimality gap can require a large number of samples in a sample average approximation, leading to large-scale instances that are computationally expensive to solve. Therefore, we develop a branch-and-cut decomposition algorithm, in which spatial branching of the first stage continuous variables and linear programming approximations for the recourse problem are sequentially tightened. We test our decomposition algorithm with multiple improvements and show it can significantly reduce solution time over solving the problem directly.

在本文中，我们考虑了一个优化问题，该问题涉及在单个中断下使活动网络崩溃。中断是一个事件，其大小和时间是随机的。当中断发生时，尚未开始或尚未完成的活动的持续时间可能会发生变化。我们制定了一个两阶段随机混合整数程序，其中阶段的时间是随机的。在我们的模型中，追索问题是一个混合整数程序。我们证明这个问题是 NP-hard 问题，并使用简单的例子来说明与问题的确定性对应物不同的属性。获得合理紧密的最优差距可能需要在样本平均近似中使用大量样本，从而导致解决计算成本高昂的大规模实例。所以，我们开发了一种分支和切割分解算法，其中第一阶段连续变量的空间分支和追索问题的线性规划近似值依次收紧。我们通过多项改进测试了我们的分解算法，并表明与直接解决问题相比，它可以显着减少求解时间。