Abstract
We consider a two-agent scheduling problem on parallel machines such that each task of agent 2 has a given time window. Furthermore, we introduce a resource constraint under which the number of simultaneously processed tasks of agent 2 is restricted, although some machines are available. The objective is to minimize the total completion time for agent 1 while the total weight of the processed tasks for agent 2 is at or above a given threshold. Because the problem is known to be strongly NP-hard, we focus on the case with unit processing time. We analyze the computational complexity for its special cases, which have some restrictions on four parameters: the weight and the duration of agent 2, the number of machines, and the maximum number of simultaneously processed tasks of agent 2.
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