Multi-scenario scheduling to maximise the weighted number of just-in-time jobs

We study a multi-scenario scheduling problem on a single-machine and a two-machine flow-shop system. The criterion is to maximise the weighted number of just-in-time jobs. We first analyze the case where only processing times are scenario-dependent. For this case, we prove that the single-machine problem is solvable in polynomial time. We also prove that the unit weight two-machine flow-shop problem is solvable in polynomial time if processing times are scenario-dependent only on the second machine, and is ordinary NP-hard when processing times are scenario-dependent only on the first machine. This ordinary NP-hard result holds as long as the number of scenarios is fixed. Otherwise, the problem becomes strongly NP-hard. We then analyze the case where only weights are scenario-dependent. We adopt a multi-criteria approach and define several problem variations. We prove that one of them is polynomial solvable on a single machine and ordinary NP-hard in a two-machine flow-shop system. We also prove that all other problem variations are ordinary NP-hard even if there are only two scenarios, and are strongly NP-hard when the number of scenarios is arbitrary. Finally, we provide two pseudo-polynomial time algorithms for solving all the hard problems when the number of scenarios is fixed.