We consider two-machine flow shop scheduling problems with storage to minimize the makespan. Under a storage constraint, each job has its own size, and the total size of the job(s) being processed on the machine(s) and the job(s) waiting for the second operation cannot exceed the given capacity. It is known that the computational complexity is strongly NP-hard and in general an optimal schedule does not exist in the set of schedules with the same job sequences on two machines, referred to as a permutation set . Thus, we consider two cases such that an optimal schedule exists in the permutation set. These two cases have the special structure of processing times. In the first case, the processing times of two machines for each job are identical, while in the second case, machine 2 dominates machine 1, that is, the smallest processing time of the second group of operations is larger than or equal to the largest one of the first group of operations. We show that the first case is strongly NP-hard and admits a polynomial-time approximation scheme (PTAS), while the second case is solved in polynomial time.

中文翻译：

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存储约束下具有最优排列时间表的两机流水车间

我们考虑使用存储的两机流水车间调度问题以最小化完工时间。在存储约束下，每个作业都有自己的大小，机器上正在处理的作业和等待第二次操作的作业的总大小不能超过给定的容量。众所周知，计算复杂度是 NP-hard 的，通常在两台机器上具有相同作业序列的调度集中不存在最优调度，称为置换集。因此，我们考虑两种情况，使得置换集中存在最优调度。这两种情况具有特殊的处理时间结构。在第一种情况下，两台机器对每个作业的处理时间是相同的，而在第二种情况下，机器 2 支配机器 1，即，第二组操作的最小处理时间大于或等于第一组操作的最大处理时间。我们表明第一种情况是强 NP-hard 并允许多项式时间近似方案（PTAS），而第二种情况在多项式时间内解决。