Sensitivity Analysis for the Single-Machine Preemptive Scheduling Problem of Minimizing Flow Time

Sensitivity analysis for the single-machine preemptive scheduling problem of minimizing flow time is discussed herein. Note that an optimal solution of the scheduling problem contains not just a combinatorial structure but also a temporal structure. The combinatorial structure specifies the sequence in which jobs or job pieces are processed. The temporal structure provides the start and completion time of every job or job piece. It is certain that a change in parameters causes a change in temporal structure. In this paper, we focus on the robust performance in the combinatorial structure, or the optimal sequence, after a single release data changes. By observing the effects of a parameter change on the optimal sequence, we found that the parameter change results only in the shifting forward (or backward) of some job blocks, and the amount of shifting forward (or backward) is equal to the amount of the first job piece size decrease (or increase) or the amount of the last job piece size increase (or decrease) for any shifting job block. Furthermore,the first job piece and the last job piece are, respectively, the first piece of the same job being processed and the last piece of the same job being processed. Therefore, to answer the question of focus in this paper, we first rewrite the optimal sequence in a job block form, and show some of its important properties. Then, a necessary and sufficient condition for the optimal sequence to remain optimal after a single release data changes is generated by four $O(nlogn)$ iterative time algorithms in the order of the job blocks.