In scheduling problems with generalized due dates (gdd), the due dates are specified according to their position in the sequence, and the j -th due date is assigned to the job in the j -th position. We study a single-machine problem with generalized due dates, where the objective is maximizing the number of jobs completed exactly on time. We prove that the problem is NP-hard in the strong sense. To our knowledge, this is the only example of a scheduling problem where the job-specific version has a polynomial-time solution, and the gdd version is strongly NP-hard. A branch-and-bound (B&B) algorithm, an integer programming (IP)-based procedure, and an efficient heuristic are introduced and tested. Both the B&B algorithm and the IP-based solution procedure can solve most medium-sized problems in a reasonable computational effort. The heuristic serves as the initial step of the B&B algorithm and in itself obtains the optimum in most cases. We also study two special cases: max-on-time for a given job sequence and max-on-time with unit-execution-time jobs. For both cases, polynomial-time dynamic programming algorithms are introduced, and large-sized problems are easily solved.

中文翻译：

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单机调度以最大化具有广义截止日期的准时作业数量

在具有广义截止日期 (gdd) 的调度问题中，截止日期是根据它们在序列中的位置指定的，并且将第 j 个截止日期分配给第 j 个位置的作业。我们研究了一个具有广义截止日期的单机问题，其目标是最大化按时完成的作业数量。我们证明了这个问题在强意义上是 NP-hard 的。据我们所知，这是调度问题的唯一示例，其中作业特定版本具有多项式时间解决方案，而 gdd 版本是强 NP 难的。引入并测试了分支定界 (B&B) 算法、基于整数规划 (IP) 的过程和有效的启发式算法。B&B 算法和基于 IP 的求解过程都可以以合理的计算量解决大多数中等规模的问题。启发式作为 B&B 算法的初始步骤，并且在大多数情况下它本身获得了最优值。我们还研究了两种特殊情况：给定作业序列的最大工作时间和具有单位执行时间工作的最大工作时间。对于这两种情况，都引入了多项式时间动态规划算法，可以轻松解决大型问题。