We study a two-machine flow shop scheduling problem where any operation can be rejected at a certain cost. A solution for such a problem requires two sets of decisions. The first involves the partition of the set of operations into two subsets: the set of operations that are accepted for scheduling in the shop, and the set of rejected operations. The second decision involves scheduling the set of accepted operations in the shop. The objective is to find a solution that minimizes the sum of the makespan and the total rejection cost. We prove that the problem is -hard even if all processing operations have identical processing times and identical rejection costs on either one of the two machines. We show, however, that the problem is fixed parameterized tractable with respect to a parameter that combine the number of different processing times on both machines with the number of different rejection costs on one out of the two machines. We also provide a pseudo-polynomial time algorithm for the problem, which we then convert into a fully polynomial time approximation scheme. This is achieved by dividing the problem into a set of subproblems and deriving a fully polynomial time approximation scheme for each one of them, separately. Finally, we present an integer linear programming formulation of the problem and two simple 2-approximation algorithms.

中文翻译：

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协调两机流水车间调度问题中的调度和拒绝决策

我们研究了两机流水车间调度问题，其中任何操作都可以以一定的成本被拒绝。此类问题的解决方案需要两组决策。第一个涉及将操作集划分为两个子集：接受车间调度的操作集和拒绝的操作集。第二个决策涉及安排车间中已接受的操作集。目标是找到一种解决方案，使完工时间和总报废成本之和最小化。我们证明，即使所有加工操作在两台机器中的任何一台上具有相同的加工时间和相同的废品成本，问题也是困难的。然而，我们表明，相对于将两台机器上的不同处理时间的数量与两台机器中的一台机器上的不同剔除成本的数量结合起来的参数，该问题是固定的参数化易处理的。我们还为该问题提供了伪多项式时间算法，然后将其转换为完全多项式时间近似方案。这是通过将问题划分为一组子问题并分别为每个子问题导出完全多项式时间近似方案来实现的。最后，我们提出了该问题的整数线性规划公式和两种简单的 2 近似算法。