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doi: 10.3934/jimo.2017057

An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time

 1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China 2 Department of Automation, Xiamen University, Xiamen, 361005, China

* Corresponding author: jipingtao@gmail.com

# These authors contributed equally to the work

Received  February 2016 Revised  September 2016 Published  June 2017

Fund Project: This work is supported by the National Natural Science Foundation of China (11501171,11571321,11201391), the Doctoral Foundation of Henan Polytechnic University (B2016-60), and the Fundamental Research Funds for the Central Universities of China (Grant No. 20720160085)

We revisit the classical online scheduling problem on parallel machines for minimizing total weighted completion time. In the problem, a set of independent jobs arriving online over time has to be scheduled on identical machines, where the information of each job including its processing time and weight is not known in advance. The goal is to minimize the total weighted completion time of the jobs. For this problem, we propose an improved 2.11-competitive online algorithm based on a kind of waiting strategy.

Citation: Ran Ma, Jiping Tao. An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2017057
References:
 [1] E. J. Anderson and C. N. Potts, Online scheduling of a single machine to minimize total weighted completion time, Mathematics of Operations Research, 29 (2004), 686-697. doi: 10.1287/moor.1040.0092. [2] M. C. Chou, M. Queyranne and D. Simchi-Levi, The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates, Mathematical Programming, 106 (2006), 137-157. doi: 10.1007/s10107-005-0588-1. [3] J. R. Correa and M. R. Wagner, LP-based online scheduling: from single to parallel machines, Mathematical Programming, 119 (2009), 109-136. doi: 10.1007/s10107-007-0204-7. [4] M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella and Y. Wang, Single machine scheduling with release dates, SIAM Journal on Discrete Mathematics, 15 (2002), 165-192. doi: 10.1137/S089548019936223X. [5] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. R. Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Annals of Discrete Mathematics, 5 (1979), 287-326. doi: 10.1016/S0167-5060(08)70356-X. [6] L. A. Hall, A. S. Schulz, D. B. Shmoys and J. Wein, Scheduling to minimize average completion time: Off-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), 513-544. doi: 10.1287/moor.22.3.513. [7] T. Kawaguchi and S. Kyan, Worst case bound of an LRF schedule for the mean weighted flow-time problem, SIAM Journal on Computing, 15 (1986), 1119-1129. doi: 10.1137/0215081. [8] P. H. Liu and X. W. Lu, On-line scheduling of parallel machines to minimize total completion times, Computers & Operations Research, 36 (2009), 2647-2652. doi: 10.1016/j.cor.2008.11.008. [9] R. Ma, J. P. Tao and J. J. Yuan, Online scheduling with linear deteriorating jobs to minimize the total weighted completion time, Applied Mathematics And Computation, 273 (2016), 570-583. doi: 10.1016/j.amc.2015.10.058. [10] R. Ma and J. J. Yuan, Online tradeoff scheduling on a single machine to minimize makespan and total weighted completion time, International Journal of Production Economics, 158 (2014), 114-119. doi: 10.1016/j.ijpe.2014.07.027. [11] N. Megow and A. S. Schulz, On-line scheduling to minimize average completion time revisited, Operations Research Letters, 32 (2004), 485-490. doi: 10.1016/j.orl.2003.11.008. [12] C. Phillips, C. Stein and J. Wein, Minimizing average completion time in the presence of release dates, Mathematical Programming, 82 (1998), 199-223. doi: 10.1007/BF01585872. [13] R. Sitters, Efficient algorithms for average completion time scheduling, in Integer Programming and Combinatorial Optimization (eds. F. Eisenbrand and F. Shepherd), vol. 6080 of Lecture Notes in Computer Science, Springer, (2010), 411-423. [14] W. Smith, Various optimizers for single-stage production problem, Naval Research Logistics Quarterly, 3 (1956), 59-66. doi: 10.1002/nav.3800030106. [15] J. P. Tao, A better online algorithm for the parallel machine scheduling to minimize the total weighted completion time, Computers & Operations Research, 43 (2014), 215-224. doi: 10.1016/j.cor.2013.09.016. [16] J. P. Tao, R. H. Huang and T. D. Liu, A 2.28-competitive algorithm for online scheduling on identical machines, Journal of Industrial and Management Optimization, 11 (2015), 185-198. doi: 10.3934/jimo.2015.11.185. [17] A. P. A. Vestjens, On-line Machine Scheduling PhD thesis, Technische Universiteit Eindhoven, Eindhoven, Netherlands, 1997.

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References:
 [1] E. J. Anderson and C. N. Potts, Online scheduling of a single machine to minimize total weighted completion time, Mathematics of Operations Research, 29 (2004), 686-697. doi: 10.1287/moor.1040.0092. [2] M. C. Chou, M. Queyranne and D. Simchi-Levi, The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates, Mathematical Programming, 106 (2006), 137-157. doi: 10.1007/s10107-005-0588-1. [3] J. R. Correa and M. R. Wagner, LP-based online scheduling: from single to parallel machines, Mathematical Programming, 119 (2009), 109-136. doi: 10.1007/s10107-007-0204-7. [4] M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella and Y. Wang, Single machine scheduling with release dates, SIAM Journal on Discrete Mathematics, 15 (2002), 165-192. doi: 10.1137/S089548019936223X. [5] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. R. Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Annals of Discrete Mathematics, 5 (1979), 287-326. doi: 10.1016/S0167-5060(08)70356-X. [6] L. A. Hall, A. S. Schulz, D. B. Shmoys and J. Wein, Scheduling to minimize average completion time: Off-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), 513-544. doi: 10.1287/moor.22.3.513. [7] T. Kawaguchi and S. Kyan, Worst case bound of an LRF schedule for the mean weighted flow-time problem, SIAM Journal on Computing, 15 (1986), 1119-1129. doi: 10.1137/0215081. [8] P. H. Liu and X. W. Lu, On-line scheduling of parallel machines to minimize total completion times, Computers & Operations Research, 36 (2009), 2647-2652. doi: 10.1016/j.cor.2008.11.008. [9] R. Ma, J. P. Tao and J. J. Yuan, Online scheduling with linear deteriorating jobs to minimize the total weighted completion time, Applied Mathematics And Computation, 273 (2016), 570-583. doi: 10.1016/j.amc.2015.10.058. [10] R. Ma and J. J. Yuan, Online tradeoff scheduling on a single machine to minimize makespan and total weighted completion time, International Journal of Production Economics, 158 (2014), 114-119. doi: 10.1016/j.ijpe.2014.07.027. [11] N. Megow and A. S. Schulz, On-line scheduling to minimize average completion time revisited, Operations Research Letters, 32 (2004), 485-490. doi: 10.1016/j.orl.2003.11.008. [12] C. Phillips, C. Stein and J. Wein, Minimizing average completion time in the presence of release dates, Mathematical Programming, 82 (1998), 199-223. doi: 10.1007/BF01585872. [13] R. Sitters, Efficient algorithms for average completion time scheduling, in Integer Programming and Combinatorial Optimization (eds. F. Eisenbrand and F. Shepherd), vol. 6080 of Lecture Notes in Computer Science, Springer, (2010), 411-423. [14] W. Smith, Various optimizers for single-stage production problem, Naval Research Logistics Quarterly, 3 (1956), 59-66. doi: 10.1002/nav.3800030106. [15] J. P. Tao, A better online algorithm for the parallel machine scheduling to minimize the total weighted completion time, Computers & Operations Research, 43 (2014), 215-224. doi: 10.1016/j.cor.2013.09.016. [16] J. P. Tao, R. H. Huang and T. D. Liu, A 2.28-competitive algorithm for online scheduling on identical machines, Journal of Industrial and Management Optimization, 11 (2015), 185-198. doi: 10.3934/jimo.2015.11.185. [17] A. P. A. Vestjens, On-line Machine Scheduling PhD thesis, Technische Universiteit Eindhoven, Eindhoven, Netherlands, 1997.
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