Rescheduling with new orders and general maximum allowable time disruptions

We study the rescheduling with new orders on a single machine under the general maximum allowable time disruptions. Under the restriction of general maximum allowable time disruptions, each original job has an upper bound for its time disruption (regarded as the maximum allowable time disruption of the job), or equivalently, in every feasible schedule, the difference of the completion time of each original job compared to that in the pre-schedule does not exceed its maximum allowable time disruption. We also consider a stronger restriction which additionally requires that, in a feasible schedule, the starting time of each original job is not allowed to be scheduled smaller than that in the pre-schedule. Scheduling objectives to be minimized are the maximum lateness and the total completion time, respectively, and the pre-schedules of original jobs are given by EDD-schedule and SPT-schedule, respectively. Then we have four problems for consideration. For the two problems for minimizing the maximum lateness, we present strong NP-hardness proof, provide a simple 2-approximation polynomial-time algorithm, and show that, unless $$\text {P}= \text {NP}$$ P = NP , the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 2. For the two problems for minimizing the total completion time, we present strong NP-hardness proof, provide a simple heuristic algorithm, and show that, unless $$\text {P}= \text {NP}$$ P = NP , the two problems cannot have an approximation polynomial-time algorithm with a performance ratio less than 4/3. Moreover, by relaxing the maximum allowable time disruptions of the original jobs, we present a super-optimal dual-approximation polynomial-time algorithm. As a consequence, if the maximum allowable time disruption of each original job is at least its processing time, then the two problems for minimizing the total completion time are solvable in polynomial time. Finally, we show that, under the agreeability assumption (i.e., the nondecreasing order of the maximum allowable time disruptions of the original jobs coincides with their scheduling order in the pre-schedule), the four problems in consideration are solvable in polynomial time.