Literature Review of Single Machine Scheduling Problem with Uniform Parallel Machines

ABSTRACT

This paper presents a survey of single machine scheduling problem with uniform parallel machines. The single machine scheduling problem with uniform parallel machines consists of n jobs, each with single operation, which are to be scheduled on m parallel machines with different speeds. These parallel machines are also called proportional machines or related machines. There are several measures of performance which are to be optimized in uniform parallel machines scheduling. Since, this scheduling problem is a combinatorial problem; usage of a heuristic is inevitable to obtain solution in polynomial time. This paper gives a classification of the literatures of this scheduling problem in three major categories, viz. offline scheduling, online scheduling and miscellaneous scheduling. In total, the available literatures are classified into 17 subgroups. Under each of the first two categories, the available literatures are discussed under different groups based on different measures of performance and non-preemptive/preemptive nature of the jobs. In the last category, the literatures are discussed under three subgroups, namely non-preemptive jobs, preemptive jobs and periodic jobs.

This paper presents a survey of single machine scheduling problem with uniform parallel machines. The single machine scheduling problem with uniform parallel machines consists of n jobs, each with single operation, which are to be scheduled on m parallel machines with different speeds. These parallel machines are also called proportional machines or related machines. There are several measures of performance which are to be optimized in uniform parallel machines scheduling. Since, this scheduling problem is a combinatorial problem; usage of a heuristic is inevitable to obtain solution in polynomial time. This paper gives a classification of the literatures of this scheduling problem in three major categories, viz. offline scheduling, online scheduling and miscellaneous scheduling. In total, the available literatures are classified into 17 subgroups. Under each of the first two categories, the available literatures are discussed under different groups based on different measures of performance and non-preemptive/preemptive nature of the jobs. In the last category, the literatures are discussed under three subgroups, namely non-preemptive jobs, preemptive jobs and periodic jobs.

Cite this paper

nullP. Senthilkumar and S. Narayanan, "Literature Review of Single Machine Scheduling Problem with Uniform Parallel Machines,"*Intelligent Information Management*, Vol. 2 No. 8, 2010, pp. 457-474. doi: 10.4236/iim.2010.28056.

nullP. Senthilkumar and S. Narayanan, "Literature Review of Single Machine Scheduling Problem with Uniform Parallel Machines,"

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[1] R. Panneerselvam, “Production and Operations Management,” 2nd Edition, Prentice-Hall of India, New Delhi, 2005.

[2] E. G. Jr. Coffman and R. L. Graham, “Optimal Scheduling for Two-Processor Systems,” Acta Informatica, Vol. 1, No. 3, 1972, pp. 200-213.

[3] E. Horowiz and S. Sahni, “Exact and Approximate Algorithms for Scheduling Nonidentical Processors,” Journal of the ACM, Vol. 23, No. 2, 1976, pp. 317-327.

[4] O. H. Ibarra and C. E. Kim, “Heuristic Algorithms for Scheduling Independent Tasks on Nonidentical Processors,” Journal of the ACM, Vol. 24, No. 2, 1977, pp. 280-289.

[5] D. Prabuddha and E. M. Thomas, “Scheduling to Minimize Makespan on Unequal Parallel Processors,” Decision Sciences, Vol. 11, No. 4, 1980, pp. 586-602.

[6] R. L. Bulfin, and R. G. Parker, “Scheduling Jobs on Two Facilities to Minimize Makespan,” Management Science, Vol. 26, No. 2, 1980, pp. 202-214.

[7] D. K. Friesen and M. A. Langston, “Bounds for Multifit Scheduling on Uniform Processors,” SIAM Journal on Computing, Vol. 12, 1983, pp. 60-70.

[8] R. L. Graham, “Bounds for Multiprocessing Timing Anomalies,” SIAM Journal of Applied Mathematics, Vol. 17, No. 2, 1969, pp. 416-429.

[9] D. Gregory, “Scheduling Independent Tasks on Uniform Processors,” SIAM Journal of Computing, Vol. 13, No. 4, 1984, pp. 705-716.

[10] D.K. Friesen, “Tighter Bounds for LPT Scheduling on Uniform Processors,” SIAM Journal on Computing, Vol. 16, No. 3, 1987, pp. 554-560.

[11] D. S. Hochbaum and D. B. Shmoys, “A Polynomial Approximation Scheme for Scheduling on Uniform Processors: Using the Dual Approximation Approach,” SIAM Journal on Computing, Vol. 17, No. 3, 1988, pp. 539-551.

[12] B. Chen, “Parametric Bounds for LPT Scheduling on Uniform Processors,” Acta Mathematicae Applicateae, Sinica, Vol. 7, No. 1, 1991, pp. 67-73.

[13] P. Mireault, J. B. Orlin and R. V. Vohra, “A Parametric Worst Case Analysis of the LPT Heuristic for Two Uniform Machines,” Operations Research, Vol. 45, No. 1, 1997, pp. 116-125.

[14] R. E. Burkard and Y. He, “A Note on MULTIFIT Scheduling for Uniform Machines,” Computing, Vol. 61, No. 3, 1998, pp. 277-283.

[15] M. Y. Kovalyov and Y. M. Shafransky, “Uniform Machine Scheduling of Unit-Time Jobs Subject to Resource Constraints,” Discrete Applied Mathematics, Vol. 84, No. 1-3, 1998, pp. 253-257.

[16] R. E. Burkard, Y. He and H. Kellerer, “A Linear Compound Algorithm for Uniform Machine Scheduling,” Computing, Vol. 61, No. 1, 1998, pp. 1-9.

[17] R. Panneerselvam and S. Kanagalingam, “Modelling Parallel Processors with Different Processing Speeds of Single Machine Scheduling Problem to Minimize Make- span,” Industrial Engineering Journal, Vol. 17, No. 6, 1998, pp. 16-19.

[18] R. Panneerselvam and S. Kanagalingam, “Simple Heuristic for Single Machine Scheduling Problem with Two Parallel Processors Having Varying Speeds to Minimize Makespan,” Industrial Engineering Journal, Vol. 18, No. 6, 1999, pp. 2-8.

[19] C. Chekuri and M. Bender, “An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines,” Proceedings of IPCO’99, LNCS, 1999, pp. 383-393.

[20] G. J. Woeginger, “A Comment on Scheduling on Uniform Machines under Chain-Type Precedence Constraints,” Operations Research Letters, Vol. 26, No. 3, 2000, pp. 107-109.

[21] F. A. Chudak and D. B. Shmoys, “An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines,” Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, 1997, pp. 581-590.

[22] J. Jaffe, “Efficient Scheduling of Tasks without Full Use of Processor Resources,” Theoretical Computer Science, Vol. 26, No. 1, 1980, pp. 22-35.

[23] C. Chekuri and M. Bender, “An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines,” Journal of Algorithms, Vol. 41, No. 2, 2001, 212-224.

[24] C. J. Liao and C.-H. Lin, “Makespan Minimization for Two Uniform Parallel Machines,” International Journal of Production Economics, Vol. 84, No. 2, 2003, pp. 205- 213.

[25] L. Epstein and J. Sgall, “Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines,” Proceedings of 7th Annual European Symposium on Algorithms, 1999, pp. 151-162.

[26] L. Epstein and J. Sgall, “Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines,” Algorithmica, Vol. 39, No. 1, 2004, pp. 43- 57.

[27] C. Koulamas and George J. Kyparisis, “Makespan Minimization on Uniform Parallel Machines with Release Times,” European Journal of Operational Research, Vol. 157, No. 1, 2004, pp. 262-266.

[28] A. Agarwal, S. Colak, V. S. Jacob and H. Pirkul, “Heuristics and Augmented Neural Networks for Task Scheduling with Non-Identical Machines,” European Journal of Operational Research, Vol. 175, No. 1, 2006, pp. 296- 317.

[29] C.-H. Lin and C.-J. Liao, “Makespan Minimization for Multiple Uniform Machines,” Computers & Industrial Engineering, Vol. 54, No. 4, 2008, pp. 983-992.

[30] T. Kis, and R. Kapolnai, “Approximation and Auctions for Scheduling Batches on Related Machines,” Operations Research Letters, Vol. 35, No. 1, 2007, pp. 61-68.

[31] S. T. McCormic and M. inedo, “Scheduling n Independent Jobs on m Uniform Machines with both Flow Time and Makespan Objectives: A Parametric Analysis,” ORSA Journal of Computing, Vol. 7, No. 1, 1995, pp. 63- 77.

[32] C. U. Martel, “A parallel Algorithm for Preemptive Scheduling of Uniform Machines,” Journal of Parallel and Distributed Computing, Vol. 5, No. 6, 1988, pp. 700- 715.

[33] T. Gonzalez and S. Sahni, “Preemptive Scheduling of Uniform Processor Systems,” Journal of Association of Computing Machinery, Vol. 25, No, 1, 1978, pp. 92-101.

[34] H. Shachnai, T. Tamir and G. J. Woeginger, “Minimizing Makespan and Preemption Costs on a System of Uniform Machines,” Proceedings of the 10th European Symposium on Algorithms, 2002, pp. 859-871.

[35] D. G. Pandelis, “Optimal Preemptive Scheduling on Uniform Machines with Discounted Flow Time Objectives,” European Journal of Operational Research, Vol. 177, No. 1, 2007, pp. 630-637.

[36] M. I. Dessouky and R. L. Marcellus, “Scheduling Identical Jobs on Uniform Parallel Machines with Random Processing Times,” Computers and Industrial Engineering, Vol. 35, No. 1-2, 1998, pp. 109-112.

[37] M. Azizoglu and O. Kirca, “On the Minimization of Total Weighted Flow Time with Identical and Uniform Parallel Machines,” European Journal of Operational Research, Vol. 113, No. 2, 1999, pp. 91-100.

[38] Z.-L. Chen and W. B. Powell, “Solving Parallel Machine Scheduling Problems by Column Generation,” INFORMS Journal on Computing, Vol. 11, 1999, pp. 78-94.

[39] T. F. Gonzalez, Y. T. L. Joseph and M. Pinedo, “Minimizing Total Completion Time on Uniform Machines with Deadline Constraints,” ACM Transactions on Algorithms, Vol. 2, No. 1, 2006, pp. 95-115.

[40] J. Y. T. Leung, H. B. Li, M. Pinedo and J. W. Zhang, “Minimizing Total Weighted Completion Time when Scheduling Orders in a Flexible Environment with Uniform Machines,” Information Processing Letters, Vol. 103, No. 3, 2007, pp. 119-129.

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