Abstract
An event in a schedule is a job start, interruption, resumption or completion time. A shift in a schedule is an non-idling interval between two events that does not contain other events. If a scheduling problem with a regular criterion has only integer data (and we consider here only such problems), then the length of smallest shifts required in optimal nonpreemptive schedules is obviously 1. The length of smallest shifts required in optimal preemptive schedules, however, can be infinitely small. As Sauer and Stone (Order 4:195–206, 1987) showed more than 25 years ago, shifts of length less than \(m^{-n}\) are not required in shortest preemptive schedules of \(n\) unit-length jobs with precedence constraints on \(m\) identical parallel machines. They showed, on the other hand, that there are instances for infinitely many \(n\) such that shifts of length less than \((m-1)^{-n/(3m)}(m-2)/m\) can be required if \(m\ge 3\). In this paper, we continue research in the same direction and strengthen their results, finding the respective tighter bounds, \(m^{-(n+1)/2}\) and \((m-1)^{-n/(m+3)}\). We also obtain similar results for other preemptive scheduling problems on identical parallel machines. A useful consequence of certain of these results is that preemptive scheduling problems with unequal release dates and/or unequal due dates can require even smaller shifts for optimality. We also identify problems whose optimal preemptive schedules do not require shifts of length less than \(1/m\).
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Notes
Although the paper of Sauer and Stone (1987) is devoted exceptionally to the study of the resolution of optimal preemptive schedules, they do not use the term “resolution” or another term instead. We find that it is appropriate and even necessary to introduce this term to name the research topic at this time. We use the term “resolution” making analogy with the image resolution, i.e., a measure of the distinguishability of details in an image, and considering a schedule chart as a special image where adjacent events must be distinguishable in a properly chosen time scale. We should also mention that Sauer and Stone use the term “shift” without a definition relying, as we think, on a common knowledge about shifts in schedules like those for medical staff in hospitals or flight attendants in the airline industry. Our definition of the term “shift” has the same meaning as one in Sauer and Stone (1987).
Events in a compact schedule cannot be made earlier by left displacements. All data specifying a rational schedule are rational.
An SPT schedule is obtained by the SPT rule which schedules jobs with shortest processing time first on available machines. It is well known that an SPT schedule of independent jobs with equal release dates is of minimum \(\sum C_j\) Graham et al. (1979).
The reviewer of this paper observed that Corollary 5 can be extended to the criteria \(\sum U_j\) and \(\sum w_jU_j\) with integer weights \(w_j\) by considering a sequence of copies of the instance \(V_{m^k}\) in which every job in one copy in this sequence precedes all jobs in the next. The number of the copies and integer \(k\) should be chosen in such a way that the length of ideal schedules for this sequence becomes integer.
Although it is not needed for the proof, we note that, in accordance with a rational structure theorem Baptiste et al. (2009b), we need to consider only the case where \(e_0,e_1,\ldots ,e_k\) are rational numbers whose sizes are polynomially bounded by the size of \(I\).
There exists a reduction from \(P|{\text {pmtn}},{\text {prec}}|L_{\max }\) to \(P|{\text {pmtn}}, {\text {prec}}',r_j|C_{\max }\), where \({\text {prec}}'\) denotes the reverse to \({\text {prec}}\) precedence constraints. Lawler et al. (1993), cf. Sect. 3.
Table 4 Lower bounds for the lengths of shifts in optimal preemptive schedules for problems in the class \(P|{\text {pmtn}},{\text {prec}},..\,|\gamma \) and related linear programs Note that the reduction based on the subdivision of jobs is always possible because job lengths in any instance of \(P|{\text {pmtn}},{\text {prec}}|C_{\max }\) can be expanded by a proper common factor such that the related instance of \(P|{\text {prec}},p_j=p|C_{\max }\) would have unit-length jobs.
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Acknowledgments
We are thankful to the anonymous reviewer who found a serious omission in the original version of this paper and also observed that Corollary 5 can be extended to the criteria of a minimum number of late jobs and a minimum weighted number of late jobs.
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The article was written when V. G. Timkovsky worked for the University of Sydney, NSW, Australia.
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Coffman, E.G., Ng, C.T. & Timkovsky, V.G. How small are shifts required in optimal preemptive schedules?. J Sched 18, 155–163 (2015). https://doi.org/10.1007/s10951-013-0355-8
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DOI: https://doi.org/10.1007/s10951-013-0355-8