On Several Scheduling Problems with Rejection or Discretely Compressible Processing Times
In the traditional scheduling problems, it is always assumed that any job has to be processed and the processing time is pre-given and fixed. In this paper, we address the scheduling problems with rejection or with discretely compressible processing times in which we can choose a subset of jobs to process or discretely compress the original processing times. Of course, choosing not to process any job or to process it with a compressed processing time incurs a corresponding penalty or cost. We consider the following problems for the first time: scheduling with discretely compressible processing times to minimize makespan with the constraint of total compression cost, scheduling with rejection to minimize the total weighted completion time with the constraint of total penalties and scheduling with discretely compressible processing times to minimize the sum of total weighted completion time plus total compression cost. We show that they are all NP-hard and design pseudo-polynomial time algorithms through dynamic programming and FPTASs for the first two problems. For the third problem, we present a greedy heuristic. Theoretical analysis shows that it has a bounded worst case performance ratio for a special case and large numbers of simulations tell us that it works very well for the general problem.
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