We study a scheduling problem involving both partitioning and scheduling decisions. A solution for our problem is defined by (i) partitioning the set of jobs into a set of accepted and a set of rejected jobs and (ii) scheduling the set of accepted jobs in a proportionate flow shop scheduling system. For a given solution, the jth largest due date is assigned to the job with the jth largest completion time. The quality of a solution is measured by two criteria, one for each set of jobs. The first is the total tardiness of the set of accepted jobs, and the second is the total rejection cost. We study two problems. The goal in the first is to find a solution minimizing the sum of the total tardiness and the rejection cost, while the goal in the second is to find a solution minimizing the total rejection cost, given a bound on the total tardiness. As both problems are \({\mathcal {N}}{\mathcal {P}}\)-hard, we focus on the design of both exact algorithms and approximation schemes.

我们研究了一个涉及分区和调度决策的调度问题。我们的问题的解决方案是这样定义的：（i）将作业集划分为接受的作业集和拒绝的作业集，以及（ii）在成比例的流水车间调度系统中调度接受的作业集。对于给定的解决方案，将第j个最大的到期日分配给具有j最大的完成时间。解决方案的质量通过两个标准来衡量，每组作业一个标准。第一个是接受作业集的总延迟，第二个是总拒绝成本。我们研究两个问题。第一个的目标是找到一个最小化总延迟和拒绝成本之和的解决方案，而第二个的目标是找到一个最小化总拒绝成本的解决方案，给定总延迟的界限。由于这两个问题都是\({\mathcal {N}}{\mathcal {P}}\) -hard，我们专注于精确算法和近似方案的设计。