带重入的流水车间：总加权完成时间目标
Flow shops with reentry: The total weighted completion time objective
AI总结：
本文研究带重入流水车间的总加权完成时间最小化问题，证明单位加工时间下问题强NP难，提出LRL和WLRL优先规则并给出性能比，同时给出机器数固定时的FPTAS和伪多项式时间算法。
摘要：
流水车间是被广泛研究的机器环境，其中所有作业必须按相同顺序访问所有机器。传统流水车间假设每个作业只穿过车间一次，但许多工业环境要求作业在完成前多次循环通过车间。这意味着作业在穿过车间并在最后一台机器上完成加工后，必须返回第一台机器并再次穿过车间，直到完成所有要求的循环。这种被称为带重入的流水车间的设置在工业中有大量应用，例如半导体制造。规划问题是在最小化总加权完成时间的同时调度所有作业的所有循环。本文考虑具有单位加工时间的重入流水车间。我们证明如果机器数量是输入的一部分，该问题是强NP难的。我们提出最少剩余循环数优先（LRL）优先规则，并证明它最小化总未加权完成时间。然后，我们分析加权最少剩余循环数优先（WLRL）优先规则，并证明其最坏情况性能比为$(1+\sqrt{2})/2$（约1.2）。此外，如果流水车间中的机器数量固定，我们提出一个完全多项式时间近似方案（FPTAS）和一个伪多项式时间算法。
英文摘要：
Flow shops are widely studied machine environments in which all jobs must visit all machines in the same order. While conventional flow shops assume that each job traverses the shop only once, many industrial environments require jobs to loop through the shop multiple times before completion. This means that after traversing the shop and completing its processing on the last machine, a job must return to the first machine and traverse the shop again until it has completed all its required loops. Such a setting, referred to as a flow shop with reentry, has numerous applications in industry, e.g., semiconductor manufacturing. The planning problem is to schedule all loops of all jobs while minimizing the total weighted completion time. In this paper, we consider reentrant flow shops with unit processing times. We show that this problem is strongly NP-hard if the number of machines is part of the input. We propose the Least Remaining Loops First (LRL) priority rule and show that it minimizes the total unweighted completion time. Then, we analyze the Weighted Least Remaining Loops First (WLRL) priority rule and show that it has a worst-case performance ratio of $(1+\sqrt{2})/2$ (about 1.2). Additionally, we present a fully polynomial time approximation scheme (FPTAS) and a pseudo-polynomial time algorithm if the number of machines in the flow shop is fixed.
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