Reactive and proactive single-machine scheduling to maintain a maximum number of starting times

This paper considers, in the single-machine scheduling context, the reactive and proactive problems arising when, due to unpredictable events between the time a baseline scheduled has been planned and the time the schedule must be implemented, the job durations may have increased so that the baseline schedule is no longer feasible. In the reactive case, the baseline schedule is known, the real job durations are known and we search for a schedule of the real instance that maximizes the number of jobs started at the same date in both schedules, this maximum being called the reactive gain. We show that, in the non-preemptive case, the corresponding decision problem is NP-complete in the strong sense while in the discrete preemptive case, it can be polynomially solved. In the proactive case, the real job durations are only known to belong to an uncertainty domain and we search for a baseline schedule that maximizes the worst reactive gain over the uncertainty domain. We show that the corresponding decision problem is NP-complete in the non-preemptive case while it is quite easy in the discrete preemptive case.