A polynomial-time approximation scheme for parallel two-stage flowshops under makespan constraint
As a hybrid of the Parallel Two-stage Flowshop problem and the Multiple Knapsack problem, we investigate the scheduling of parallel two-stage flowshops under makespan constraint, which was motivated by applications in cloud computing and introduced by Chen et al. [3] recently. A set of two-stage job...
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| Published in: | Theoretical computer science Vol. 922; pp. 438 - 446 |
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| Language: | English |
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24.06.2022
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | As a hybrid of the Parallel Two-stage Flowshop problem and the Multiple Knapsack problem, we investigate the scheduling of parallel two-stage flowshops under makespan constraint, which was motivated by applications in cloud computing and introduced by Chen et al. [3] recently. A set of two-stage jobs are selected and scheduled on parallel two-stage flowshops to achieve the maximum total profit while maintaining the given makespan constraint. We give a positive answer to an open question about its approximability proposed by Chen et al. [3]. More specifically, based on guessing strategies and rounding techniques for linear programs, we present a polynomial time approximation scheme (PTAS) for the case when the number of flowshops is a fixed constant. As the case with two flowshops is already strongly NP-hard, our PTAS achieves the best possible approximation ratio. |
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| AbstractList | As a hybrid of the Parallel Two-stage Flowshop problem and the Multiple Knapsack problem, we investigate the scheduling of parallel two-stage flowshops under makespan constraint, which was motivated by applications in cloud computing and introduced by Chen et al. [3] recently. A set of two-stage jobs are selected and scheduled on parallel two-stage flowshops to achieve the maximum total profit while maintaining the given makespan constraint. We give a positive answer to an open question about its approximability proposed by Chen et al. [3]. More specifically, based on guessing strategies and rounding techniques for linear programs, we present a polynomial time approximation scheme (PTAS) for the case when the number of flowshops is a fixed constant. As the case with two flowshops is already strongly NP-hard, our PTAS achieves the best possible approximation ratio. |
| Author | Tong, Weitian Xu, Yao Zhang, Huili |
| Author_xml | – sequence: 1 givenname: Weitian orcidid: 0000-0002-9815-2330 surname: Tong fullname: Tong, Weitian email: wtong.research@gmail.com organization: Department of Computer Science, Georgia Southern University, Statesboro, GA 30458, USA – sequence: 2 givenname: Yao surname: Xu fullname: Xu, Yao email: yxu@georgiasouthern.edu organization: Department of Computer Science, Georgia Southern University, Statesboro, GA 30458, USA – sequence: 3 givenname: Huili surname: Zhang fullname: Zhang, Huili email: zhang.huilims@xjtu.edu.cn organization: School of Management, Xi'an Jiaotong University; and State Key Lab for Manufacturing Systems Engineering, Xi'an 710049, China |
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| Keywords | Multiple knapsacks Rounding Polynomial-time approximation scheme Makespan constraint Parallel two-stage flowshops |
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| References | Jansen (br0100) 2010; 39 Ibarra, Kim (br0090) 1975; 22 Lau, Ravi, Singh (br0150) 2011 Dawande, Gavirneni, Rachamadugu (br0040) 2006; 44 Potra, Wright (br0160) 2000; 124 Chen, Huang, Guo (br0030) 2021 He, Kusiak, Artiba (br0080) 1996; 28 Kellerer (br0130) 1999 Tong, Miyano, Goebel, Lin (br0180) 2018; 734 Wu, Chen, Wang (br0230) 2020; 818 Wu, Chen, Wang (br0220) 2020; 806 Rennie, Dobson (br0170) 1969; 7 Dong, Jin, Luo, Tong (br0050) 2020; 281 Dong, Tong, Luo, Wang, Hu, Xu, Lin (br0060) 2017; 657 Blazewicz, Ecker, Pesch, Schmidt, Weglarz (br0010) 2019 Johnson (br0120) 1954; 1 Zhang, Velde (br0240) 2012; 216 Chekuri, Khanna (br0020) 2005; 35 Kovalyov (br0140) 1985 Wu, Chen, Wang (br0200) 2019; 786 Garey, Johnson (br0070) 1979 Vairaktarakis, Elhafsi (br0190) 2000; 32 Jansen (br0110) 2012 Wu, Chen, Wang (br0210) 2019; 776 Jansen (10.1016/j.tcs.2022.04.044_br0110) 2012 Vairaktarakis (10.1016/j.tcs.2022.04.044_br0190) 2000; 32 Dawande (10.1016/j.tcs.2022.04.044_br0040) 2006; 44 Dong (10.1016/j.tcs.2022.04.044_br0050) 2020; 281 Rennie (10.1016/j.tcs.2022.04.044_br0170) 1969; 7 Johnson (10.1016/j.tcs.2022.04.044_br0120) 1954; 1 Chen (10.1016/j.tcs.2022.04.044_br0030) 2021 Kellerer (10.1016/j.tcs.2022.04.044_br0130) 1999 Lau (10.1016/j.tcs.2022.04.044_br0150) 2011 Tong (10.1016/j.tcs.2022.04.044_br0180) 2018; 734 Wu (10.1016/j.tcs.2022.04.044_br0210) 2019; 776 Dong (10.1016/j.tcs.2022.04.044_br0060) 2017; 657 Ibarra (10.1016/j.tcs.2022.04.044_br0090) 1975; 22 Wu (10.1016/j.tcs.2022.04.044_br0230) 2020; 818 Zhang (10.1016/j.tcs.2022.04.044_br0240) 2012; 216 Wu (10.1016/j.tcs.2022.04.044_br0220) 2020; 806 Garey (10.1016/j.tcs.2022.04.044_br0070) 1979 Kovalyov (10.1016/j.tcs.2022.04.044_br0140) 1985 Blazewicz (10.1016/j.tcs.2022.04.044_br0010) 2019 Potra (10.1016/j.tcs.2022.04.044_br0160) 2000; 124 He (10.1016/j.tcs.2022.04.044_br0080) 1996; 28 Jansen (10.1016/j.tcs.2022.04.044_br0100) 2010; 39 Chekuri (10.1016/j.tcs.2022.04.044_br0020) 2005; 35 Wu (10.1016/j.tcs.2022.04.044_br0200) 2019; 786 |
| References_xml | – volume: 39 start-page: 1392 year: 2010 end-page: 1412 ident: br0100 article-title: Parameterized approximation scheme for the multiple knapsack problem publication-title: SIAM J. Comput. – volume: 35 start-page: 713 year: 2005 end-page: 728 ident: br0020 article-title: A polynomial time approximation scheme for the multiple knapsack problem publication-title: SIAM J. Comput. – year: 1979 ident: br0070 article-title: Computers and Intractability, vol. 174 – volume: 776 start-page: 117 year: 2019 end-page: 124 ident: br0210 article-title: Scheduling two-stage jobs on multiple flowshops publication-title: Theor. Comput. Sci. – volume: 216 start-page: 544 year: 2012 end-page: 552 ident: br0240 article-title: Approximation algorithms for the parallel flow shop problem publication-title: Eur. J. Oper. Res. – start-page: 83 year: 2021 end-page: 95 ident: br0030 article-title: Scheduling on multiple two-stage flowshops with a deadline publication-title: AAIM – volume: 28 start-page: 129 year: 1996 end-page: 139 ident: br0080 article-title: A scheduling problem in glass manufacturing publication-title: IIE Trans. – start-page: 51 year: 1999 end-page: 62 ident: br0130 article-title: A polynomial time approximation scheme for the multiple knapsack problem publication-title: Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques – volume: 734 start-page: 24 year: 2018 end-page: 31 ident: br0180 article-title: An approximation scheme for minimizing the makespan of the parallel identical multi-stage flow-shops publication-title: Theor. Comput. Sci. – volume: 1 start-page: 61 year: 1954 end-page: 68 ident: br0120 article-title: Optimal two- and three-stage production schedules with setup times included publication-title: Nav. Res. Logist. Q. – year: 2019 ident: br0010 article-title: Handbook on Scheduling – year: 1985 ident: br0140 article-title: Efficient epsilon-approximation algorithm for minimizing the makespan in a parallel two-stage system publication-title: Vesti Academii Navuk Belaruskai SSR. Seria Phizika-Matematichnikh Navuk – year: 2011 ident: br0150 article-title: Iterative Methods in Combinatorial Optimization, vol. 46 – volume: 7 start-page: 116 year: 1969 end-page: 121 ident: br0170 article-title: On Stirling numbers of the second kind publication-title: J. Comb. Theory – volume: 806 start-page: 509 year: 2020 end-page: 515 ident: br0220 article-title: Improved approximation algorithms for two-stage flowshops scheduling problem publication-title: Theor. Comput. Sci. – volume: 22 start-page: 463 year: 1975 end-page: 468 ident: br0090 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: J. ACM – volume: 281 start-page: 16 year: 2020 end-page: 24 ident: br0050 article-title: A polynomial-time approximation scheme for an arbitrary number of parallel two-stage flow-shops publication-title: Eur. J. Oper. Res. – volume: 124 start-page: 281 year: 2000 end-page: 302 ident: br0160 article-title: Interior-point methods publication-title: J. Comput. Appl. Math. – volume: 786 start-page: 67 year: 2019 end-page: 77 ident: br0200 article-title: On scheduling inclined jobs on multiple two-stage flowshops publication-title: Theor. Comput. Sci. – start-page: 313 year: 2012 end-page: 324 ident: br0110 article-title: A fast approximation scheme for the multiple knapsack problem publication-title: International Conference on Current Trends in Theory and Practice of Computer Science – volume: 32 start-page: 687 year: 2000 end-page: 699 ident: br0190 article-title: The use of flowlines to simplify routing complexity in two-stage flowshops publication-title: IIE Trans. – volume: 818 start-page: 74 year: 2020 end-page: 82 ident: br0230 article-title: On scheduling multiple two-stage flowshops publication-title: Theor. Comput. Sci. – volume: 44 start-page: 73 year: 2006 end-page: 84 ident: br0040 article-title: Scheduling a two-stage flowshop under makespan constraint publication-title: Math. Comput. Model. – volume: 657 start-page: 64 year: 2017 end-page: 72 ident: br0060 article-title: An fptas for the parallel two-stage flowshop problem publication-title: Theor. Comput. Sci. – volume: 35 start-page: 713 issue: 3 year: 2005 ident: 10.1016/j.tcs.2022.04.044_br0020 article-title: A polynomial time approximation scheme for the multiple knapsack problem publication-title: SIAM J. Comput. doi: 10.1137/S0097539700382820 – start-page: 313 year: 2012 ident: 10.1016/j.tcs.2022.04.044_br0110 article-title: A fast approximation scheme for the multiple knapsack problem – volume: 657 start-page: 64 year: 2017 ident: 10.1016/j.tcs.2022.04.044_br0060 article-title: An fptas for the parallel two-stage flowshop problem publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2016.04.046 – volume: 44 start-page: 73 issue: 1–2 year: 2006 ident: 10.1016/j.tcs.2022.04.044_br0040 article-title: Scheduling a two-stage flowshop under makespan constraint publication-title: Math. Comput. Model. doi: 10.1016/j.mcm.2004.12.016 – volume: 7 start-page: 116 issue: 2 year: 1969 ident: 10.1016/j.tcs.2022.04.044_br0170 article-title: On Stirling numbers of the second kind publication-title: J. Comb. Theory doi: 10.1016/S0021-9800(69)80045-1 – volume: 818 start-page: 74 year: 2020 ident: 10.1016/j.tcs.2022.04.044_br0230 article-title: On scheduling multiple two-stage flowshops publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2018.04.017 – volume: 22 start-page: 463 issue: 4 year: 1975 ident: 10.1016/j.tcs.2022.04.044_br0090 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: J. ACM doi: 10.1145/321906.321909 – year: 1979 ident: 10.1016/j.tcs.2022.04.044_br0070 – volume: 1 start-page: 61 issue: 1 year: 1954 ident: 10.1016/j.tcs.2022.04.044_br0120 article-title: Optimal two- and three-stage production schedules with setup times included publication-title: Nav. Res. Logist. Q. doi: 10.1002/nav.3800010110 – volume: 734 start-page: 24 year: 2018 ident: 10.1016/j.tcs.2022.04.044_br0180 article-title: An approximation scheme for minimizing the makespan of the parallel identical multi-stage flow-shops publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2017.09.018 – start-page: 83 year: 2021 ident: 10.1016/j.tcs.2022.04.044_br0030 article-title: Scheduling on multiple two-stage flowshops with a deadline – volume: 776 start-page: 117 year: 2019 ident: 10.1016/j.tcs.2022.04.044_br0210 article-title: Scheduling two-stage jobs on multiple flowshops publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2019.01.017 – volume: 786 start-page: 67 year: 2019 ident: 10.1016/j.tcs.2022.04.044_br0200 article-title: On scheduling inclined jobs on multiple two-stage flowshops publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2018.04.005 – volume: 39 start-page: 1392 issue: 4 year: 2010 ident: 10.1016/j.tcs.2022.04.044_br0100 article-title: Parameterized approximation scheme for the multiple knapsack problem publication-title: SIAM J. Comput. doi: 10.1137/080731207 – year: 2011 ident: 10.1016/j.tcs.2022.04.044_br0150 – volume: 28 start-page: 129 year: 1996 ident: 10.1016/j.tcs.2022.04.044_br0080 article-title: A scheduling problem in glass manufacturing publication-title: IIE Trans. doi: 10.1080/07408179608966258 – volume: 806 start-page: 509 year: 2020 ident: 10.1016/j.tcs.2022.04.044_br0220 article-title: Improved approximation algorithms for two-stage flowshops scheduling problem publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2019.08.028 – volume: 32 start-page: 687 year: 2000 ident: 10.1016/j.tcs.2022.04.044_br0190 article-title: The use of flowlines to simplify routing complexity in two-stage flowshops publication-title: IIE Trans. doi: 10.1080/07408170008967427 – year: 2019 ident: 10.1016/j.tcs.2022.04.044_br0010 – start-page: 51 year: 1999 ident: 10.1016/j.tcs.2022.04.044_br0130 article-title: A polynomial time approximation scheme for the multiple knapsack problem – volume: 124 start-page: 281 issue: 1–2 year: 2000 ident: 10.1016/j.tcs.2022.04.044_br0160 article-title: Interior-point methods publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(00)00433-7 – volume: 216 start-page: 544 year: 2012 ident: 10.1016/j.tcs.2022.04.044_br0240 article-title: Approximation algorithms for the parallel flow shop problem publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2011.08.007 – volume: 281 start-page: 16 issue: 1 year: 2020 ident: 10.1016/j.tcs.2022.04.044_br0050 article-title: A polynomial-time approximation scheme for an arbitrary number of parallel two-stage flow-shops publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2019.08.019 – year: 1985 ident: 10.1016/j.tcs.2022.04.044_br0140 article-title: Efficient epsilon-approximation algorithm for minimizing the makespan in a parallel two-stage system |
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| SubjectTerms | Makespan constraint Multiple knapsacks Parallel two-stage flowshops Polynomial-time approximation scheme Rounding |
| Title | A polynomial-time approximation scheme for parallel two-stage flowshops under makespan constraint |
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