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November  2022, 18(6): 3989-4010. doi: 10.3934/jimo.2021144

A best possible algorithm for an online scheduling problem with position-based learning effect

Ran Ma 1,, , Lu Zhang 1, and  Yuzhong Zhang 2,

1. 

School of Management Engineering, , Qingdao University of Technology, Qingdao 266525, China, University Research Center for Smart City Construction and Management of Shandong Province, Qingdao 266525, China
2. 

Institute of Operations Research, School of Management, Qufu Normal University, Rizhao 276826, China

*Corresponding author: Ran Ma

Received  January 2021 Revised  April 2021 Published  November 2022 Early access  September 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Grant Nos. 11501171, 11771251) and the Province Natural Science Foundation of Shandong (Grant No. ZR2020MA028)

In this paper, we focus on an online scheduling problem with position-based learning effect on a single machine, where the jobs are released online over time and preemption is not allowed. The information about each job $ J_j $, including the basic processing time $ p_j $ and the release time $ r_j $, is only available when it arrives. The actual processing time $ p_j' $ of each job $ J_j $ is defined as a function related to its position $ r $, i.e., $ p_j' = p_j(\alpha-r\beta) $, where $ \alpha $ and $ \beta $ are both nonnegative learning index. Our goal is to minimize the sum of completion time of all jobs. For this problem, we design a deterministic polynomial time online algorithm Delayed Shortest Basic Processing Time (DSBPT). In order to facilitate the understanding of the online algorithm, we present a relatively common and simple example to describe the execution process of the algorithm, and then by competitive analysis, we show that online algorithm DSBPT is a best possible online algorithm with a competitive ratio of 2.

Keywords: scheduling, online algorithm, competitive ratio, single machine, learning effect.
Mathematics Subject Classification: Primary: 90B35.
Citation: Ran Ma, Lu Zhang, Yuzhong Zhang. A best possible algorithm for an online scheduling problem with position-based learning effect. Journal of Industrial and Management Optimization, 2022, 18 (6) : 3989-4010. doi: 10.3934/jimo.2021144
References:
[1]

W. Allihaibi, M. Cholette, M. Masoud, J. Burke and A. Karim, A heuristic approach for scheduling patient treatment in an emergency department based on bed blocking, International Journal of Industrial Engineering Computations, 11 (2020), 565-584.  doi: 10.5267/j.ijiec.2020.4.005.

[2]

D. Y. Bai, H. Y. Xue, L. Wang, C. C. Wu, W. C. Lin and D. H. Abdulkadir, Effective algorithms for single-machine learning-effect scheduling to minimize completion-time-based criteria with release dates, Expert Systems With Applications, 156 (2020), 113445.

[3]

L. Bai, D. Yang, X. Wang, L. Tong, X. Zhu, N. Zhong, C. Bai and C. A. Powell, Chinese experts consensus on the Internet of Things-aided diagnosis and treatment of coronavirus disease 2019 (COVID-19), Clinical eHealth, 3 (2020), 7-15.  doi: 10.1016/j.ceh.2020.03.001.

[4]

D. Biskup, Single-machine scheduling with learning considerations, European J. Oper. Res., 115 (1999), 173-178.  doi: 10.1016/S0377-2217(98)00246-X.

[5]

D. Biskup, A state-of-the-art review on scheduling with learning effects, European J. Oper. Res., 188 (2008), 315-329.  doi: 10.1016/j.ejor.2007.05.040.

[6]

T. C. E. Cheng, S. R. Cheng, W. H. Wu, P. H. Hsu and C. C. Wu, A two-agent single machine scheduling problem with truncated sum-of-processing-times-based learning considerations, Computers and Industrial Engineering, 60 (2011), 534-541. 

[7]

M. B. Cheng, S. X. Xiao and G. Liu, Single-machine rescheduling problems with learning effect under disruptions, J. Ind. Manag. Optim., 14 (2018), 967-980.  doi: 10.3934/jimo.2017085.

[8]

J. A. Hoogeveen and A. P. A. Vestjens, Optimal on-line algorithms for single-machine scheduling, Lecture Notes in Comput. Sci., Springer, Berlin, (1996), 404–414.

[9]

Z. Y. Jiang, F. F. Chen and X. D. Zhang, Single-machine scheduling with times-based and job-dependent learning effect, Journal of the Operational Research Society, 68 (2017), 809-815. 

[10]

S. Jun and S. Lee, Learning dispatching rules for single machine scheduling with dynamic arrivals based on decision trees and feature construction, International Journal of Production Research, 59 (2020), 2838-2856.  doi: 10.1080/00207543.2020.1741716.

[11]

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy and D. B. Shmoys, Sequencing and scheduling algorithms and complexity, Handbooks in Operations Research and Management Science, 4 (1993), 445-522. 

[12]

W. C. Lee and C. C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model., 33 (2009), 2159-2163.  doi: 10.1016/j.apm.2008.05.020.

[13]

P. H. Liu and X. W. Lu, On-line scheduling of parallel machines to minimize total completion times, Comput. Oper. Res., 36 (2009), 2647-2652.  doi: 10.1016/j.cor.2008.11.008.

[14]

Y. Y. Lu, F. Teng and Z. X. Feng, Scheduling jobs with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects, Asia-Pac. J. Oper. Res., 32 (2015), 1550026. doi: 10.1142/S0217595915500268.

[15]

R. Ma and J. P. Tao, An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time, J. Ind. Manag. Optim., 14 (2018), 497-510.  doi: 10.3934/jimo.2017057.

[16]

R. Ma and S. N. Guo, Applying "Peeling Onion" approach for competitive analysis in online scheduling with rejection, European J. Oper. Res., 290 (2021), 57-67.  doi: 10.1016/j.ejor.2020.08.009.

[17]

G. Mosheiov, Scheduling problems with a learning effect, European J. Oper. Res., 132 (2001), 687-693.  doi: 10.1016/S0377-2217(00)00175-2.

[18]

G. Mosheiov, Minimizing total absolute deviation of job completion times: Extensions to position-dependent processing times and parallel identical machines, J. Oper. Res. Soc., 59 (2008), 1422-1424.  doi: 10.1057/palgrave.jors.2602480.

[19]

S. Mustu and T. Eren, The single machine scheduling problem with setup times under an extension of the general learning and forgetting effects, Optim. Lett., 15 (2021), 1327-1343.  doi: 10.1007/s11590-020-01641-9.

[20]

J. B. Wang, M. Gao, J. J. Wang, L. Liu and H. Y. He, Scheduling with a position-weighted learning effect and job release dates, Eng. Optim., 52 (2019), 1475-1493.  doi: 10.1080/0305215X.2019.1664498.

[21]

J.-B. Wang, L. H. Sun and L. Y. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Appl. Math. Model., 34 (2010), 2813-2819.  doi: 10.1016/j.apm.2009.12.015.

[22]

J.-B. Wang and J.-J. Wang, Single machine scheduling with sum-of-logarithm processing-times based and position based learning effects, Optim. Lett., 8 (2014), 971-982.  doi: 10.1007/s11590-012-0494-4.

[23]

J. B. Wang and Z. Q. Xia, Flow-shop scheduling with a learning effect, J. Oper. Res. Soc., 56 (2005), 1325-1330.  doi: 10.1057/palgrave.jors.2601856.

[24]

T. P. Wright, Factors affecting the cost of airplanes, J. Aer. Sci., 3 (1936), 122-128.  doi: 10.2514/8.155.

[25]

S.-J. Yang and D.-L. Yang, Note on "A note on single-machine group scheduling problems with position-based learning effect", Appl. Math. Model., 34 (2010), 4306-4308.  doi: 10.1016/j.apm.2010.03.037.

[26]

A. E. Zade, S. S. Haghighi and M. Soltani, Reinforcement learning for optimal scheduling of Glioblastoma treatment with Temozolomide, Computer Methods and Programs in Biomedicine, 193 (2020), 105443.

show all references

References:
[1]

W. Allihaibi, M. Cholette, M. Masoud, J. Burke and A. Karim, A heuristic approach for scheduling patient treatment in an emergency department based on bed blocking, International Journal of Industrial Engineering Computations, 11 (2020), 565-584.  doi: 10.5267/j.ijiec.2020.4.005.

[2]

D. Y. Bai, H. Y. Xue, L. Wang, C. C. Wu, W. C. Lin and D. H. Abdulkadir, Effective algorithms for single-machine learning-effect scheduling to minimize completion-time-based criteria with release dates, Expert Systems With Applications, 156 (2020), 113445.

[3]

L. Bai, D. Yang, X. Wang, L. Tong, X. Zhu, N. Zhong, C. Bai and C. A. Powell, Chinese experts consensus on the Internet of Things-aided diagnosis and treatment of coronavirus disease 2019 (COVID-19), Clinical eHealth, 3 (2020), 7-15.  doi: 10.1016/j.ceh.2020.03.001.

[4]

D. Biskup, Single-machine scheduling with learning considerations, European J. Oper. Res., 115 (1999), 173-178.  doi: 10.1016/S0377-2217(98)00246-X.

[5]

D. Biskup, A state-of-the-art review on scheduling with learning effects, European J. Oper. Res., 188 (2008), 315-329.  doi: 10.1016/j.ejor.2007.05.040.

[6]

T. C. E. Cheng, S. R. Cheng, W. H. Wu, P. H. Hsu and C. C. Wu, A two-agent single machine scheduling problem with truncated sum-of-processing-times-based learning considerations, Computers and Industrial Engineering, 60 (2011), 534-541. 

[7]

M. B. Cheng, S. X. Xiao and G. Liu, Single-machine rescheduling problems with learning effect under disruptions, J. Ind. Manag. Optim., 14 (2018), 967-980.  doi: 10.3934/jimo.2017085.

[8]

J. A. Hoogeveen and A. P. A. Vestjens, Optimal on-line algorithms for single-machine scheduling, Lecture Notes in Comput. Sci., Springer, Berlin, (1996), 404–414.

[9]

Z. Y. Jiang, F. F. Chen and X. D. Zhang, Single-machine scheduling with times-based and job-dependent learning effect, Journal of the Operational Research Society, 68 (2017), 809-815. 

[10]

S. Jun and S. Lee, Learning dispatching rules for single machine scheduling with dynamic arrivals based on decision trees and feature construction, International Journal of Production Research, 59 (2020), 2838-2856.  doi: 10.1080/00207543.2020.1741716.

[11]

E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy and D. B. Shmoys, Sequencing and scheduling algorithms and complexity, Handbooks in Operations Research and Management Science, 4 (1993), 445-522. 

[12]

W. C. Lee and C. C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model., 33 (2009), 2159-2163.  doi: 10.1016/j.apm.2008.05.020.

[13]

P. H. Liu and X. W. Lu, On-line scheduling of parallel machines to minimize total completion times, Comput. Oper. Res., 36 (2009), 2647-2652.  doi: 10.1016/j.cor.2008.11.008.

[14]

Y. Y. Lu, F. Teng and Z. X. Feng, Scheduling jobs with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects, Asia-Pac. J. Oper. Res., 32 (2015), 1550026. doi: 10.1142/S0217595915500268.

[15]

R. Ma and J. P. Tao, An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time, J. Ind. Manag. Optim., 14 (2018), 497-510.  doi: 10.3934/jimo.2017057.

[16]

R. Ma and S. N. Guo, Applying "Peeling Onion" approach for competitive analysis in online scheduling with rejection, European J. Oper. Res., 290 (2021), 57-67.  doi: 10.1016/j.ejor.2020.08.009.

[17]

G. Mosheiov, Scheduling problems with a learning effect, European J. Oper. Res., 132 (2001), 687-693.  doi: 10.1016/S0377-2217(00)00175-2.

[18]

G. Mosheiov, Minimizing total absolute deviation of job completion times: Extensions to position-dependent processing times and parallel identical machines, J. Oper. Res. Soc., 59 (2008), 1422-1424.  doi: 10.1057/palgrave.jors.2602480.

[19]

S. Mustu and T. Eren, The single machine scheduling problem with setup times under an extension of the general learning and forgetting effects, Optim. Lett., 15 (2021), 1327-1343.  doi: 10.1007/s11590-020-01641-9.

[20]

J. B. Wang, M. Gao, J. J. Wang, L. Liu and H. Y. He, Scheduling with a position-weighted learning effect and job release dates, Eng. Optim., 52 (2019), 1475-1493.  doi: 10.1080/0305215X.2019.1664498.

[21]

J.-B. Wang, L. H. Sun and L. Y. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Appl. Math. Model., 34 (2010), 2813-2819.  doi: 10.1016/j.apm.2009.12.015.

[22]

J.-B. Wang and J.-J. Wang, Single machine scheduling with sum-of-logarithm processing-times based and position based learning effects, Optim. Lett., 8 (2014), 971-982.  doi: 10.1007/s11590-012-0494-4.

[23]

J. B. Wang and Z. Q. Xia, Flow-shop scheduling with a learning effect, J. Oper. Res. Soc., 56 (2005), 1325-1330.  doi: 10.1057/palgrave.jors.2601856.

[24]

T. P. Wright, Factors affecting the cost of airplanes, J. Aer. Sci., 3 (1936), 122-128.  doi: 10.2514/8.155.

[25]

S.-J. Yang and D.-L. Yang, Note on "A note on single-machine group scheduling problems with position-based learning effect", Appl. Math. Model., 34 (2010), 4306-4308.  doi: 10.1016/j.apm.2010.03.037.

[26]

A. E. Zade, S. S. Haghighi and M. Soltani, Reinforcement learning for optimal scheduling of Glioblastoma treatment with Temozolomide, Computer Methods and Programs in Biomedicine, 193 (2020), 105443.

Figure 1.  The scheduling of DSBPT and an off-line optimal scheduling
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Figure 2.  Block and subblock
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Figure 3.  Reverse pair
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Table 1.  Notations-1
Notation Meaning
$ J_j $ the job of index $ j $, where $ j=1,2,\ldots,n $
$ I $ a job instance, $ I=\left\{J_1,J_2,\ldots,J_n\right\} $
$ r_j(I) $ the release time of job $ J_j $ in $ I $
$ p_j(I) $ the basic processing time of job $ J_j $ in $ I $
$ p'_j(I) $ the actual processing time of job $ J_j $ in $ I $
$ X(I) $ a feasible schedule of $ I $
$ S_j(X(I)) $ the starting time of job $ J_j $ in $ X(I) $
$ C_j(X(I)) $ the completion time of job $ J_j $ in $ X(I) $,
$ C_j(X(I))=S_j(X(I))+p_j(I) $
$ Z_j(X(I)) $ the contribution of job $ J_j $ to the total objective value of $ I $ in $ X(I) $
$ Z(X(I)) $ the total objective value of $ I $ in $ X(I) $,
$ Z(X(I))=\sum Z_j(X(I)) $
$ \pi(I) $ an off-line optimal schedule of $ I $
$ {\rm{OPT}}( I )$ the total objective value of $ I $ in $ \pi(I) $, $ {\rm{OPT}}(I)=\sum Z_j(\pi(I)) $
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Notation Meaning
$ J_j $ the job of index $ j $, where $ j=1,2,\ldots,n $
$ I $ a job instance, $ I=\left\{J_1,J_2,\ldots,J_n\right\} $
$ r_j(I) $ the release time of job $ J_j $ in $ I $
$ p_j(I) $ the basic processing time of job $ J_j $ in $ I $
$ p'_j(I) $ the actual processing time of job $ J_j $ in $ I $
$ X(I) $ a feasible schedule of $ I $
$ S_j(X(I)) $ the starting time of job $ J_j $ in $ X(I) $
$ C_j(X(I)) $ the completion time of job $ J_j $ in $ X(I) $,
$ C_j(X(I))=S_j(X(I))+p_j(I) $
$ Z_j(X(I)) $ the contribution of job $ J_j $ to the total objective value of $ I $ in $ X(I) $
$ Z(X(I)) $ the total objective value of $ I $ in $ X(I) $,
$ Z(X(I))=\sum Z_j(X(I)) $
$ \pi(I) $ an off-line optimal schedule of $ I $
$ {\rm{OPT}}( I )$ the total objective value of $ I $ in $ \pi(I) $, $ {\rm{OPT}}(I)=\sum Z_j(\pi(I)) $
Table 2.  Notations-2
Notation Meaning
$ |T| $ the number of jobs in $ T $
$ T(X(I)) $ the set of jobs of $ T $ in $ X(I) $
$ Z_T(X(I)) $ the total contribution of jobs of $ T $ to $ Z(X(I)) $,
$ Z_T(X(I))=\sum_{J_j\in T} Z_j(X(I)) $
$ {\rm{OPT}}(T) $ the total objective value of jobs of $ T $ in an off-line optimal schedule, $ {\rm{OPT}}(T)=\sum Z_j(\pi(T)) $
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Notation Meaning
$ |T| $ the number of jobs in $ T $
$ T(X(I)) $ the set of jobs of $ T $ in $ X(I) $
$ Z_T(X(I)) $ the total contribution of jobs of $ T $ to $ Z(X(I)) $,
$ Z_T(X(I))=\sum_{J_j\in T} Z_j(X(I)) $
$ {\rm{OPT}}(T) $ the total objective value of jobs of $ T $ in an off-line optimal schedule, $ {\rm{OPT}}(T)=\sum Z_j(\pi(T)) $
Algorithm1 
Algorithm 1 DSBPT
At each decision time $ t $, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are many such valid jobs, the job $ J_j $ with the earliest release time is selected, presume its processing position is $ k $ if $ p_j(I)(\alpha-k\beta)\leq t $, and process the job; otherwise the machine will remain idle and wait until the next decision time $ t $ to make a decision.
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Algorithm 1 DSBPT
At each decision time $ t $, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are many such valid jobs, the job $ J_j $ with the earliest release time is selected, presume its processing position is $ k $ if $ p_j(I)(\alpha-k\beta)\leq t $, and process the job; otherwise the machine will remain idle and wait until the next decision time $ t $ to make a decision.
Algorithm2 
Algorithm 1: Online Algorithm DSBPT.
1: Input: job instance $ I $ ($ |I|\geq 1 $) and learning index $ \alpha $ and $ \beta $
2: $ S $ is an empty sequence, $ k\leftarrow 1 $, $ C\leftarrow 0 $, $ A $ is an available jobs set, $ f(S)\leftarrow 0 $, $ dt\leftarrow 0 $, $ Ct\leftarrow 0 $, $ t\leftarrow \min\{r_j|J_j\in I\} $
3: While True do
4:       update available jobs set A, $ dt\leftarrow \max\{t,Ct\} $
5:       $ D \leftarrow \{J_j\in A|\ the \ job \ has \ the \ shortest \ processing \ time \ first $
            $ \qquad \qquad \qquad \qquad \ and \ then \ releases \ the \ earliest\} $
6:       arbitrarily select a job $ J_a $ from $ D $
7:       If $ dt \geq p_a(\alpha-k\beta) $ then
8:          $ S\leftarrow S + \{J_a\} $
9:          $ C\leftarrow dt+p_a(\alpha-k\beta) $
10:          $ f(S)\leftarrow f(S)+C $
11:          $ k\leftarrow k+1 $
12:       end if
13:       If $ C>dt $ then
14:             $ Ct\leftarrow C $
15:       end if
16: end while
17: return $ S $ and $ f(S) $
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Algorithm 1: Online Algorithm DSBPT.
1: Input: job instance $ I $ ($ |I|\geq 1 $) and learning index $ \alpha $ and $ \beta $
2: $ S $ is an empty sequence, $ k\leftarrow 1 $, $ C\leftarrow 0 $, $ A $ is an available jobs set, $ f(S)\leftarrow 0 $, $ dt\leftarrow 0 $, $ Ct\leftarrow 0 $, $ t\leftarrow \min\{r_j|J_j\in I\} $
3: While True do
4:       update available jobs set A, $ dt\leftarrow \max\{t,Ct\} $
5:       $ D \leftarrow \{J_j\in A|\ the \ job \ has \ the \ shortest \ processing \ time \ first $
            $ \qquad \qquad \qquad \qquad \ and \ then \ releases \ the \ earliest\} $
6:       arbitrarily select a job $ J_a $ from $ D $
7:       If $ dt \geq p_a(\alpha-k\beta) $ then
8:          $ S\leftarrow S + \{J_a\} $
9:          $ C\leftarrow dt+p_a(\alpha-k\beta) $
10:          $ f(S)\leftarrow f(S)+C $
11:          $ k\leftarrow k+1 $
12:       end if
13:       If $ C>dt $ then
14:             $ Ct\leftarrow C $
15:       end if
16: end while
17: return $ S $ and $ f(S) $
Table 3.  Information of instance $ I $
$ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $
release time 0 8 17 23 28 30
basic processing time 8 6 12 10 4 8
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$ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $
release time 0 8 17 23 28 30
basic processing time 8 6 12 10 4 8
Table 4.  The selection process of jobs under $ \sigma(I) $
decision time valid job set selected job sequence of finished jobs current objective function value
0 $ J_1 $ None None 0
8 $ J_1 $ $ J_2 $ $ J_2 $ None 0
14 $ J_1 $ $ J_1 $ $ J_2 $ 14
17 $ J_3 $ None $ J_2 $ 14
20.8 $ J_3 $ $ J_3 $ $ J_2 $-$ J_1 $ 34.8
23 $ J_4 $ None $ J_2 $-$ J_1 $ 34.8
28 $ J_4 $ $ J_5 $ None $ J_2 $-$ J_1 $ 34.8
29.2 $ J_4 $ $ J_5 $ $ J_5 $ $ J_2 $-$ J_1 $-$ J_3 $ 64
30 $ J_4 $ $ J_6 $ None $ J_2 $-$ J_1 $-$ J_3 $ 64
31.4 $ J_4 $ $ J_6 $ $ J_6 $ $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $ 95.4
34.6 $ J_4 $ $ J_4 $ $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $ 130
37.1 None None $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $-$ J_4 $ 167.1
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decision time valid job set selected job sequence of finished jobs current objective function value
0 $ J_1 $ None None 0
8 $ J_1 $ $ J_2 $ $ J_2 $ None 0
14 $ J_1 $ $ J_1 $ $ J_2 $ 14
17 $ J_3 $ None $ J_2 $ 14
20.8 $ J_3 $ $ J_3 $ $ J_2 $-$ J_1 $ 34.8
23 $ J_4 $ None $ J_2 $-$ J_1 $ 34.8
28 $ J_4 $ $ J_5 $ None $ J_2 $-$ J_1 $ 34.8
29.2 $ J_4 $ $ J_5 $ $ J_5 $ $ J_2 $-$ J_1 $-$ J_3 $ 64
30 $ J_4 $ $ J_6 $ None $ J_2 $-$ J_1 $-$ J_3 $ 64
31.4 $ J_4 $ $ J_6 $ $ J_6 $ $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $ 95.4
34.6 $ J_4 $ $ J_4 $ $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $ 130
37.1 None None $ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $-$ J_4 $ 167.1
Table 5.  The selection process of jobs under $ \pi(I) $
time valid job set selected job sequence of finished jobs current objective function value
0 $ J_1 $ $ J_1 $ None 0
8 $ J_2 $ $ J_2 $ $ J_1 $ 8
13.1 None None $ J_1 $-$ J_2 $ 21.1
17 $ J_3 $ $ J_3 $ $ J_1 $-$ J_2 $ 21.1
23 $ J_4 $ None $ J_1 $-$ J_2 $ 21.1
25.4 $ J_4 $ $ J_4 $ $ J_1 $-$ J_2 $-$ J_3 $ 46.5
28 $ J_5 $ None $ J_1 $-$ J_2 $-$ J_3 $ 46.5
30 $ J_5 $ $ J_6 $ None $ J_1 $-$ J_2 $-$ J_3 $ 46.5
30.9 $ J_5 $ $ J_6 $ $ J_5 $ $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $ 77.4
32.5 $ J_6 $ $ J_6 $ $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $ 109.9
34.5 None None $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $-$ J_6 $ 144.4
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time valid job set selected job sequence of finished jobs current objective function value
0 $ J_1 $ $ J_1 $ None 0
8 $ J_2 $ $ J_2 $ $ J_1 $ 8
13.1 None None $ J_1 $-$ J_2 $ 21.1
17 $ J_3 $ $ J_3 $ $ J_1 $-$ J_2 $ 21.1
23 $ J_4 $ None $ J_1 $-$ J_2 $ 21.1
25.4 $ J_4 $ $ J_4 $ $ J_1 $-$ J_2 $-$ J_3 $ 46.5
28 $ J_5 $ None $ J_1 $-$ J_2 $-$ J_3 $ 46.5
30 $ J_5 $ $ J_6 $ None $ J_1 $-$ J_2 $-$ J_3 $ 46.5
30.9 $ J_5 $ $ J_6 $ $ J_5 $ $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $ 77.4
32.5 $ J_6 $ $ J_6 $ $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $ 109.9
34.5 None None $ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $-$ J_6 $ 144.4
Table 6.  The comparison between $ \sigma(I) $ and $ \pi(I) $
job processing scheduling objective function value
$ \sigma(I) $ $ J_2-J_1-J_3-J_5-J_6-J_4 $ 167.1
$ \pi(I) $ $ J_1-J_2-J_3-J_4-J_5-J_6 $ 144.4
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job processing scheduling objective function value
$ \sigma(I) $ $ J_2-J_1-J_3-J_5-J_6-J_4 $ 167.1
$ \pi(I) $ $ J_1-J_2-J_3-J_4-J_5-J_6 $ 144.4
Algorithm3 
Algorithm 2 FDSBPT
At arbitrary time $ t $, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are multiple jobs with the shortest processing time, choose arbitrary job $ J_j $ flexibly, presume its processing position is k, if $ p_j(I)(\alpha-k\beta)\leq t $, we arrange job $ J_j $ on the idle machine at time $ t $; otherwise, we do nothing until the next time.
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Algorithm 2 FDSBPT
At arbitrary time $ t $, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are multiple jobs with the shortest processing time, choose arbitrary job $ J_j $ flexibly, presume its processing position is k, if $ p_j(I)(\alpha-k\beta)\leq t $, we arrange job $ J_j $ on the idle machine at time $ t $; otherwise, we do nothing until the next time.
Algorithm4 
Algorithm 2: Online Algorithm DSBPT.
1: Input: job instance $ I $ ($ |I|\geq 1 $) and learning index $ \alpha $ and $ \beta $
2: $ S $ is an empty sequence, $ k\leftarrow 1 $, $ C\leftarrow 0 $, $ A $ is an available jobs set, $ f(S)\leftarrow 0 $, $ t\leftarrow 0 $
3: While True do
4:       update available jobs set A
5:       $ D \leftarrow \{ {J_j} \in A|\;the\;job\;has\;the\;shortest\;processing\;time{\rm{\} }} $
6:       arbitrarily select a job $ J_a $ from $ D $
7:       If $ t \geq p_a(\alpha-k\beta) $ then
8:          $ S\leftarrow S + \{J_a\} $
9:          $ C\leftarrow t+p_a(\alpha-k\beta) $
10:          $ f(S)\leftarrow f(S)+C $
11:          $ k\leftarrow k+1 $
12:       end if
13:       $ t\leftarrow t+1$
14: end while
15: return $ S $ and $ f(S) $
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Algorithm 2: Online Algorithm DSBPT.
1: Input: job instance $ I $ ($ |I|\geq 1 $) and learning index $ \alpha $ and $ \beta $
2: $ S $ is an empty sequence, $ k\leftarrow 1 $, $ C\leftarrow 0 $, $ A $ is an available jobs set, $ f(S)\leftarrow 0 $, $ t\leftarrow 0 $
3: While True do
4:       update available jobs set A
5:       $ D \leftarrow \{ {J_j} \in A|\;the\;job\;has\;the\;shortest\;processing\;time{\rm{\} }} $
6:       arbitrarily select a job $ J_a $ from $ D $
7:       If $ t \geq p_a(\alpha-k\beta) $ then
8:          $ S\leftarrow S + \{J_a\} $
9:          $ C\leftarrow t+p_a(\alpha-k\beta) $
10:          $ f(S)\leftarrow f(S)+C $
11:          $ k\leftarrow k+1 $
12:       end if
13:       $ t\leftarrow t+1$
14: end while
15: return $ S $ and $ f(S) $
Table 7.  Information of instance $ I_c $
$ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $ $ J_8 $ $ J_9 $ $ J_{10} $
release time 0 8 17 23 28 30 30 33 35 39
basic processing time 8 6 12 10 4 8 12 9 10 11
$ J_{11} $ $ J_{12} $ $ J_{13} $ $ J_{14} $ $ J_{15} $ $ J_{16} $ $ J_{17} $ $ J_{18} $ $ J_{19} $ $ J_{20} $
release time 41 42 46 48 48 48 48 50 52 52
basic processing time 13 6 7 9 8 8 9 5 13 8
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$ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $ $ J_8 $ $ J_9 $ $ J_{10} $
release time 0 8 17 23 28 30 30 33 35 39
basic processing time 8 6 12 10 4 8 12 9 10 11
$ J_{11} $ $ J_{12} $ $ J_{13} $ $ J_{14} $ $ J_{15} $ $ J_{16} $ $ J_{17} $ $ J_{18} $ $ J_{19} $ $ J_{20} $
release time 41 42 46 48 48 48 48 50 52 52
basic processing time 13 6 7 9 8 8 9 5 13 8
Table 8.  The selection process of jobs under $\sigma(I_c)$
decision time valid job set selected job current objective function value
0 $J_1$ None 0
8 $J_1$ $J_2$ $J_2$ 0
14 $J_1$ $J_1$ 14
17 $J_3$ None 14
21.84 $J_3$ $J_3$ 35.84
23 $J_4$ None 35.84
28 $J_4$ $J_5$ None 35.84
30 $J_4$ $J_5$ $J_6$ $J_7$ None 35.84
33 $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ None 35.84
33.36 $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_5$ 69.2
35 $J_4$ $J_6$ $J_7$ $J_8$ $J_9$ None 69.2
37.12 $J_4$ $J_6$ $J_7$ $J_8$ $J_9$ $J_6$ 106.32
39 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ None 106.32
41 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ None 106.32
42 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ None 106.32
44.48 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ $J_{12}$ 150.8
46 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ None 150.8
48 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ None 150.8
49.88 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{13}$ 200.68
50 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ None 200.68
52 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ None 200.68
56.04 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ $J_{18}$ 256.72
60.34 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{15}$ 317.06
67.06 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{16}$ 384.12
73.62 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{20}$ 457.74
80.02 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_8$ 537.76
87.04 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{14}$ 624.8
93.88 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$ $J_{17}$ 718.68
100.54 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_4$ 819.22
107.74 $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_9$ 926.96
114.74 $J_7$ $J_{10}$ $J_{11}$ $J_{19}$ $J_{10}$ 1041.7
122.22 $J_7$ $J_{11}$ $J_{19}$ $J_7$ 1163.92
130.14 $J_{11}$ $J_{19}$ $J_{11}$ 1294.06
138.46 $J_{19}$ $J_{19}$ 1432.52
146.52 None None 1579.04
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decision time valid job set selected job current objective function value
0 $J_1$ None 0
8 $J_1$ $J_2$ $J_2$ 0
14 $J_1$ $J_1$ 14
17 $J_3$ None 14
21.84 $J_3$ $J_3$ 35.84
23 $J_4$ None 35.84
28 $J_4$ $J_5$ None 35.84
30 $J_4$ $J_5$ $J_6$ $J_7$ None 35.84
33 $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ None 35.84
33.36 $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_5$ 69.2
35 $J_4$ $J_6$ $J_7$ $J_8$ $J_9$ None 69.2
37.12 $J_4$ $J_6$ $J_7$ $J_8$ $J_9$ $J_6$ 106.32
39 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ None 106.32
41 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ None 106.32
42 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ None 106.32
44.48 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ $J_{12}$ 150.8
46 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ None 150.8
48 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ None 150.8
49.88 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{13}$ 200.68
50 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ None 200.68
52 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ None 200.68
56.04 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ $J_{18}$ 256.72
60.34 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{15}$ 317.06
67.06 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{16}$ 384.12
73.62 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{20}$ 457.74
80.02 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_8$ 537.76
87.04 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{14}$ 624.8
93.88 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$ $J_{17}$ 718.68
100.54 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_4$ 819.22
107.74 $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_9$ 926.96
114.74 $J_7$ $J_{10}$ $J_{11}$ $J_{19}$ $J_{10}$ 1041.7
122.22 $J_7$ $J_{11}$ $J_{19}$ $J_7$ 1163.92
130.14 $J_{11}$ $J_{19}$ $J_{11}$ 1294.06
138.46 $J_{19}$ $J_{19}$ 1432.52
146.52 None None 1579.04
Table 9.  The selection process of jobs under π(Ic)
time valid job set selected job current objective function value
0 $J_1$ $J_1$ 0
8 $J_2$ $J_2$ 8
13.88 None None 21.88
17 $J_3$ $J_3$ 21.88
28.52 $J_4$ $J_5$ $J_5$ 50.4
32.28 $J_4$ $J_6$ $J_7$ $J_6$ 82.68
39.64 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_8$ 122.32
47.74 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ $J_{13}$ $J_{12}$ 170.06
53.02 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ $J_{18}$ 223.08
57.32 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{13}$ 280.4
63.2 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{15}$ 343.6
69.76 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{16}$ 413.36
76.16 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{20}$ 489.52
82.4 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{14}$ 571.92
89.24 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$ $J_{17}$ 661.16
95.9 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_4$ 757.06
103.1 $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_9$ 860.16
110.1 $J_7$ $J_{10}$ $J_{11}$ $J_{19}$ $J_{10}$ 970.26
117.58 $J_7$ $J_{11}$ $J_{19}$ $J_7$ 1087.84
125.5 $J_{11}$ $J_{19}$ $J_{11}$ 1213.34
133.82 $J_{19}$ $J_{19}$ 1347.16
141.88 None None 1489.04
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time valid job set selected job current objective function value
0 $J_1$ $J_1$ 0
8 $J_2$ $J_2$ 8
13.88 None None 21.88
17 $J_3$ $J_3$ 21.88
28.52 $J_4$ $J_5$ $J_5$ 50.4
32.28 $J_4$ $J_6$ $J_7$ $J_6$ 82.68
39.64 $J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_8$ 122.32
47.74 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ $J_{13}$ $J_{12}$ 170.06
53.02 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$ $J_{18}$ 223.08
57.32 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{13}$ 280.4
63.2 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{15}$ 343.6
69.76 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{16}$ 413.36
76.16 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$ $J_{20}$ 489.52
82.4 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{14}$ 571.92
89.24 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$ $J_{17}$ 661.16
95.9 $J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_4$ 757.06
103.1 $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$ $J_9$ 860.16
110.1 $J_7$ $J_{10}$ $J_{11}$ $J_{19}$ $J_{10}$ 970.26
117.58 $J_7$ $J_{11}$ $J_{19}$ $J_7$ 1087.84
125.5 $J_{11}$ $J_{19}$ $J_{11}$ 1213.34
133.82 $J_{19}$ $J_{19}$ 1347.16
141.88 None None 1489.04
Table 10.  The comparison between $ \sigma(I_c) $ and $ \pi(I_c) $
job processing scheduling objective function value
$ \sigma(I_c) $ $ J_2-J_1-J_3-J_5-J_6-J_{12}-J{13}-J_{18}-J_{15}-J_{16}-J_{20}-J_8-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $ 1579.04
$ \pi(I_c) $ $ J_1-J_2-J_3-J_5-J_6-J_8-J_{12}-J_{18}-J{13}-J_{15}-J_{16}-J_{20}-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $ 1489.04
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job processing scheduling objective function value
$ \sigma(I_c) $ $ J_2-J_1-J_3-J_5-J_6-J_{12}-J{13}-J_{18}-J_{15}-J_{16}-J_{20}-J_8-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $ 1579.04
$ \pi(I_c) $ $ J_1-J_2-J_3-J_5-J_6-J_8-J_{12}-J_{18}-J{13}-J_{15}-J_{16}-J_{20}-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $ 1489.04
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