We study a problem of scheduling n jobs on machines of two types: in‐house machines and third‐party machines. Scheduling on in‐house machines incurs no additional costs, while using third‐party machines implies costs depending on their number and the time of usage. Each job has a fixed time interval for being processed which can be divided and allocated among several machines, as long as there is only one machine processing the job at any time. No machine can process more than one job at a time. Jobs can be rejected, and they are of different importance that is reflected in the weight of each job. The objective is to find a subset of the jobs and the number of third‐party machines for any period of time so that the accepted jobs can be feasibly scheduled, the total weight of the accepted jobs is maximized, and the total machine usage costs does not exceed a given upper bound. We also study a similar problem in which the objective is to maximize the total time at which at least one job is processed. Both problems are encountered in situations in which certain activities with given start and completion times have to be serviced by human operators. Examples are air traffic control and the monitoring safe vehicle unloading. Other examples are the employment of subcontractors in agriculture, construction or transportation. We will present NP‐hardness proofs, polynomial and pseudo‐polynomial optimal algorithms and an approximation algorithm for these problems and their special cases. These problems admit graph‐theoretical interpretations associated with finding independent sets and a proper vertex coloring in interval graphs.

中文翻译：

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使用第三方机器进行固定间隔调度

我们研究调度问题ñ两种类型的计算机上的作业：内部计算机和第三方计算机。在内部计算机上进行调度不会产生任何额外成本，而使用第三方计算机则意味着取决于其数量和使用时间的成本。每个作业都有一个固定的处理时间间隔，只要在任何时候只有一台机器处理该作业，就可以在几台机器之间分配和分配该时间间隔。没有机器一次可以处理一项以上的工作。作业可以被拒绝，并且它们的重要性不同，这反映在每个作业的权重上。目的是找到一段时间内作业的子集和第三方计算机的数量，以便可以合理地安排接受的作业，最大化接受的作业的总权重，并且总的机器使用成本不超过给定的上限。我们还研究了一个类似的问题，其目的是最大化处理至少一项作业的总时间。在必须由人工操作员维修具有给定开始和完成时间的某些活动的情况下会遇到两个问题。例子包括空中交通管制和监视安全的车辆卸载。其他例子是在农业，建筑业或运输业中雇用分包商。我们将介绍NP硬度证明，多项式和伪多项式最佳算法以及这些问题及其特殊情况的近似算法。这些问题使图形理论上的解释与在间隔图中找到独立的集合和适当的顶点着色有关。