We show by induction that the shortest processing time sequence is optimal for a number of three-stage ordered flow shops with either synchronous flow, blocking or no-idle machines, effectively proving the optimality of an index priority rule when the adjacent job pairwise interchange argument does not hold. We also show that when the middle machine is maximal, the synchronous flow, blocking and no-wait problems are equivalent, because they can be effectively decomposed into two equivalent two-stage problems. A similar equivalence is shown for the classical flow shop and the flow shop with no-idle machines. These equivalences facilitate the solution of one problem by using the optimal algorithm for the equivalent problem. Finally, we observe that when the middle machine is minimal, the optimal sequence is not a pyramid sequence for the synchronous flow and blocking flow shops. On the other hand, we show that the optimal sequence for the flow shop with no-idle machines is a pyramid sequence obtainable by dynamic programming in pseudo-polynomial time.

中文翻译：

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具有同步流、阻塞或无空闲机器的三级有序流水车间

我们通过归纳证明最短处理时间序列对于具有同步流、阻塞或无空闲机器的多个三级有序流水车间是最优的，有效地证明了当相邻作业成对交换参数时索引优先级规则的最优性不成立。我们还表明，当中间机器最大时，同步流、阻塞和无等待问题是等价的，因为它们可以有效地分解为两个等价的两阶段问题。对于经典的流水车间和带有无空闲机器的流水车间，显示了类似的等效性。这些等价通过对等价问题使用最优算法来促进解决一个问题。最后，我们观察到当中间机器最小时，对于同步流和阻塞流车间，最佳序列不是金字塔序列。另一方面，我们表明无空闲机器的流水车间的最佳序列是一个金字塔序列，可通过伪多项式时间的动态规划获得。