We study flow shop scheduling with stochastic reentry, where jobs must complete multiple passes through the entire shop, and the number of passes that a job requires for completion is drawn from a discrete probability distribution. The goal is to find policies that minimize performance measures in expectation. Our main contribution is a reduction to a classical parallel machine scheduling problem augmented with machine arrivals. This reduction preserves expected objective values and enables transferring structural results and performance guarantees from the auxiliary problems to the reentrant flow shop setting. We demonstrate the usefulness of this reduction by proving the optimality of simple priority policies for minimizing the makespan and the total completion time in expectation under geometric and, more generally, monotone hazard rate distributions. For minimizing the total weighted completion time, we derive an approximation guarantee that depends only on the squared coefficient of variation of the underlying distributions for a simple priority policy. Our results constitute the first optimality and approximation guarantees for flow shops with stochastic reentry and demonstrate that established scheduling policies naturally extend to this setting through the proposed reduction.

翻译：我们研究具有随机重入特性的流水车间调度问题，其中每项作业需多次通过整个车间，且作业完成所需通过次数服从离散概率分布。本文目标是寻找使期望性能指标最小化的调度策略。主要贡献在于将该问题归约为带机器到达的经典并行机调度问题。该归约过程保持了期望目标值，使得能够将辅助问题的结构特性与性能保证迁移至重入流水车间场景。通过证明几何分布及更一般的单调风险率分布下，简单优先级策略在最小化期望最大完工时间与总完工时间方面的最优性，我们验证了该归约的有效性。对于最小化期望加权总完工时间，我们推导出简单优先级策略的近似比仅取决于底层分布的变异系数平方。本文研究结果为具有随机重入特性的流水车间调度提供了首个最优性与近似性保证，并表明通过所提出的归约方法，经典调度策略可自然扩展至该场景。