Abstract
We consider a single-machine scheduling problem with two competing agents. The objective is to minimize the total weighted completion time of jobs of agent 1 with a constraint on the total weight of the just-in-time jobs of agent 2. Our problem can be categorized into the non-preemptive and the preemptive versions, depending on whether the preemption is allowed for jobs of agent 1. First, we show that some open cases of the first version are NP-hard. Then, we categorize the second version into several cases, depending on whether the processing times or the weights of each agent are identical. We analyze how the computational complexity is affected by the identical processing times or the identical weights of each agent.
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