Abstract
In scheduling problems with generalized due dates (gdd), the due dates are specified according to their position in the sequence, and the j-th due date is assigned to the job in the j-th position. We study a single-machine problem with generalized due dates, where the objective is maximizing the number of jobs completed exactly on time. We prove that the problem is NP-hard in the strong sense. To our knowledge, this is the only example of a scheduling problem where the job-specific version has a polynomial-time solution, and the gdd version is strongly NP-hard. A branch-and-bound (B&B) algorithm, an integer programming (IP)-based procedure, and an efficient heuristic are introduced and tested. Both the B&B algorithm and the IP-based solution procedure can solve most medium-sized problems in a reasonable computational effort. The heuristic serves as the initial step of the B&B algorithm and in itself obtains the optimum in most cases. We also study two special cases: max-on-time for a given job sequence and max-on-time with unit-execution-time jobs. For both cases, polynomial-time dynamic programming algorithms are introduced, and large-sized problems are easily solved.
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Acknowledgements
This research was supported by the ISRAEL SCIENCE FOUNDATION (Grant No. 2505/19). The second author was also supported by the Charles I. Rosen Chair of Management and by The Recanati Fund of The School of Business Administration, The Hebrew University, Jerusalem, Israel.
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Appendix A: problems Sets for the numerical tests
Appendix A: problems Sets for the numerical tests
Problem Set 1:
Number of jobs: \( n = 10, 11, 12, 13, 14, 15. \)
Job processing times were generated uniformly in the interval \( \left[ {1,50} \right]. \)
Due dates were generated uniformly in the interval \( \left[ {1,Q} \right], \) where \( Q = \alpha \mathop \sum \limits_{j = 1}^{n} p_{j} . \)
Tightness factor: \( \alpha = 0.75, 1.0, 1.25. \)
Problem Set 2:
Number of jobs: \( n = 100. \)
Job processing times were generated uniformly in the interval \( \left[ {1,100} \right]. \)
Due dates were generated uniformly in the interval \( \left[ {1,Q} \right], \) where \( Q = \alpha \mathop \sum \limits_{j = 1}^{n} p_{j} . \)
Tightness factor: \( \alpha = 0.75, 1.0, 1.25. \)
Problem Set 3:
Number of jobs: \( n = 10, 20, \ldots , 70 \)
Job processing times were generated uniformly in the interval \( \left[ {1,50} \right]. \)
Due dates were generated uniformly in the interval \( \left[ {1,Q} \right], \) where \( Q = \alpha \mathop \sum \limits_{j = 1}^{n} p_{j} . \)
Tightness factor: \( \alpha = 0.75, 1.0, 1.25. \)
Problem Set 4:
Number of jobs: \( n = 100, 200, \ldots ,1000. \)
Job processing times were generated uniformly in the interval \( \left[ {1,50} \right]. \) Due dates were generated uniformly in the interval \( \left[ {1,Q} \right], \) where \( Q = \alpha \mathop \sum \limits_{j = 1}^{n} p_{j} . \)
Tightness factor: \( \alpha = 0.75, 1.0, 1.25. \)
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Gerstl, E., Mosheiov, G. Single machine scheduling to maximize the number of on-time jobs with generalized due-dates. J Sched 23, 289–299 (2020). https://doi.org/10.1007/s10951-020-00638-7
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DOI: https://doi.org/10.1007/s10951-020-00638-7