Two-agent single-machine scheduling problem with just-in-time jobs

We consider two scheduling problems on a single machine with two agents. The objective is to optimize the performance measure for agent 1 while maintaining the weighted number of just-in-time jobs for agent 2 at or above a given threshold. The performance measures for agent 1 are the total weighted completion times and the weighted number of tardy jobs. We analyze how the complexities of the two problems change, depending on the conditions of processing times and weights.