In this work we consider a problem of scheduling preemptable, independent jobs, characterized by the fact that their processing speeds depend on the amounts of a continuous, renewable resource allocated to jobs at a time. Jobs are scheduled on parallel, identical machines, with the criterion of minimization of the schedule length. Since two categories of resources occur in the problem: discrete (set of machines) and continuous, it is generally called a discrete-continuous scheduling problem. The model studied in this paper allows the total available amount of the continuous resource to vary over time, which is a practically important generalization that has not been considered yet for discrete-continuous scheduling problems. For this model we give some properties of optimal schedules on a basis of which we propose a general methodology for solving the considered class of problems. The methodology uses a two-phase approach in which, firstly, an assignment of machines to jobs is defined and, secondly, for this assignment an optimal continuous resource allocation is found by solving an appropriate mathematical programming problem. In the approach various cases are considered, following from assumptions made on the form of the processing speed functions of jobs. For each case an iterative algorithm is designed, leading to an optimal solution in a finite number of steps.
@article{bwmeta1.element.bwnjournal-article-amcv26i3p693bwm,
author = {Rafa\l\ R\'o\.zycki and Grzegorz Walig\'ora and Jan W\k eglarz},
title = {Scheduling preemptable jobs on identical processors under varying availability of an additional continuous resource},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {26},
year = {2016},
pages = {693-706},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv26i3p693bwm}
}
Rafał Różycki; Grzegorz Waligóra; Jan Węglarz. Scheduling preemptable jobs on identical processors under varying availability of an additional continuous resource. International Journal of Applied Mathematics and Computer Science, Tome 26 (2016) pp. 693-706. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv26i3p693bwm/
[000] Błażewicz, J., Ecker, K., Schmidt, G., Pesch, E. and Węglarz, J. (2007). Handbook of Scheduling: From Theory to Applications, Springer, Berlin. | Zbl 1165.90009
[001] Gorczyca, M. and Janiak, A. (2010). Resource level minimization in the discrete-continuous scheduling, European Journal of Operational Research 203(1): 32-41. | Zbl 1176.90216
[002] Janiak, A. (1991). Single machine scheduling problem with a common deadline and resource dependent release dates, European Journal of Operational Research 53(3): 317-325. | Zbl 0743.90066
[003] Karmarkar, N.K. (1984). A new polynomial time algorithm for linear programming, Combinatorica 4(4): 373-395. | Zbl 0557.90065
[004] Kis, T. (2005). A branch-and-cut algorithm for scheduling of projects with variable-intensity activities, Mathematical Programming 103(3): 515-139. | Zbl 1125.90019
[005] Leachman, R.C. (1983). Multiple resource leveling in construction systems through variation of activity intensities, Naval Research Logistics Quarterly 30(3): 187-198. | Zbl 0529.90061
[006] Leachman, R.C., Dincerler, A. and Kim, S. (1990). Resource-constrained scheduling of projects with variable-intensity activities, IIE Transactions 22(1): 31-39.
[007] Różycki, R. and Węglarz, J. (2012). Power-aware scheduling of preemptable jobs on identical parallel processors to meet deadlines, European Journal of Operational Research 218(1): 68-75. | Zbl 1244.90101
[008] Waligóra, G. (2011). Heuristic approaches to discrete-continuous project scheduling problems to minimize the makespan, Computational Optimization and Applications 48(2): 399-421. | Zbl 1219.90070
[009] Waligóra, G. (2014). Discrete-continuous project scheduling with discounted cash inflows and various payment models-a review of recent results, Annals of Operations Research 213(1): 319-340. | Zbl 1296.90061
[010] Węglarz, J. (1976). Time-optimal control of resource allocation in a complex of operations framework, IEEE Transactions on Systems, Man and Cybernetics 6(11): 783-788. | Zbl 0336.49022