Optimal interval scheduling with nonidentical given machines
- 41 Downloads
Abstract
We consider an interval scheduling problem where n jobs are required to be carried out by m nonidentical machines in an offline-scheduling way. Each job has a starting time, a finishing time and a number of processing units. Every machine has different number of processing units to carry out jobs. A machine can process only one job at a time without interrupted on the condition that the number of its units must satisfy job’s requirement. Further more, all units in one machine consume energy if the machine is powered up. Within this setting, one is asked to find a proper schedule of machines so that the total number of working units is as less as possible. For this interval scheduling problem, we first discuss an exact method based on integer programming which can be solved by branch-and-bound algorithm. Then, we propose two approximated methods named GreedyBS and GreedyMR using greedy strategy. GreedyBS is proved to be a 2.1343-approximated algorithm. All proposed algorithms are tested on a large set of randomly generated instances. It turns out that GreedyBS requires less total units of machines under time constrain when comparing with GreedyMR and exact methods in most cases, while GreedyMR costs the minimum time. Several parameters of GreedyBS and GreedyMR were also evaluated to improve the performances of these two algorithms.
Keywords
Scheduling Exact algorithms Greedy algorithms Performance evaluationNotes
Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant No. 71701207), Natural Science Foundation of Hunan Province(Grant No. 2018JJ2477) and the National Defense Science and Technology Project Fund of the Central military commission (Grant No. 3101097).
References
- 1.Kolen, A.W.J., Lenstra, J.K., Papadimitriou, C.H., Spieksma, F.C.R.: Interval scheduling: a survey. Naval Res. Logist. 54(5), 530–543 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Jansen, K.: An approximation algorithm for the license and shift class design problem. Eur. J. Oper. Res. 73(1), 127–131 (1994)CrossRefzbMATHGoogle Scholar
- 3.Kroon, L.G., Salomon, M., Van Wassenhove, L.N.: Exact and approximation algorithms for the tactical fixed interval scheduling problem. Eur. J. Oper. Res. 82(4), 624–638 (1995)MathSciNetzbMATHGoogle Scholar
- 4.Jansen, K.: Approximation Results for the Optimum Cost Chromatic Partition Problem. Springer, Berlin, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
- 5.Mhring, R.H.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, Cambridge (1980)Google Scholar
- 6.Bhatia, R., Chuzhoy, J., Freund, A., Naor, J.S.: Algorithmic aspects of bandwidth trading. ASM Trans. Algorithms 2719(1), 193–193 (2002)zbMATHGoogle Scholar
- 7.Huang, Q., Lloyd, E.: Cost constrained fixed job scheduling. In: Lecture Notes in Computer Science, vol. 2841, pp. 111–124 (2003)Google Scholar
- 8.Angelelli, E., Bianchessi, N., Filippi, C.: Optimal interval scheduling with a resource constraint. Comput. Oper. Res. 51(3), 268–281 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Krumke, S.O., Thielen, C., Westphal, S.: Interval scheduling on related machines. Comput. Oper. Res. 38(12), 1836–1844 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Fung, S.P.Y., Poon, C.K., Zheng, F.: Improved randomized online scheduling of intervals and jobs. Theory Comput. Syst. 55(1), 202–228 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Branda, M., Novotn, J., Olstad, A.: Fixed interval scheduling under uncertainty a tabu search algorithm for an extended robust coloring formulation. Comput. Ind. Eng. 93, 45–54 (2015)CrossRefGoogle Scholar
- 12.Angelelli, E., Filippi, C.: On the complexity of interval scheduling with a resource constraint. Theor. Comput. Sci. 412(29), 3650–3657 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Gavruskin, A., Khoussainov, B., Kokho, M., Liu, J.: Dynamic algorithms for monotonic interval scheduling problem. Theor. Comput. Sci. 562(C), 227–242 (2014)MathSciNetzbMATHGoogle Scholar
- 14.Kovalyov, M.Y., Ng, C.T., Edwin Cheng, T.C.: Fixed interval scheduling: models, applications, computational complexity and algorithms. Eur. J. Oper. Res. 178(2), 331–342 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions—I. Math. Program. 14(1), 265–294 (1978)CrossRefzbMATHGoogle Scholar