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Numerical method for solving optimal control problems with phase constraints
A hybrid metaheuristic algorithm to minimize the number of tardy jobs in a dynamic twomachine flow shop problem
1.  Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran 
2.  Department of Industrial and Systems Engineering, Isfahan University of Technology, 8415683111, Isfahan, Iran 
In this paper, first, the problem of dynamic twomachine flow shop scheduling with the objective of minimizing the number of tardy jobs is investigated. Second, a hybrid algorithm based on genetic algorithm and parallel simulated annealing algorithm is presented. In order to solve large scale instances of the problem and to generate the initial population for the hybrid approach, a heuristic algorithm is also presented. To evaluate the efficiency of the proposed algorithms, they are compared with an optimal branchandbound algorithm which has been already developed in the literature. Computational experiments demonstrate that the proposed hybrid algorithm can solve the entire smallsized problems and more than 95% of mediumsized problems optimally.
References:
[1] 
M. Abouei Ardakan, A. Hakimian, M. T. Rezvan, A branchandbound algorithm for minimising the number of tardy jobs in a twomachine flowshop problem with release datest, International Journal of Computer Integrated Manufacturing, 27 (2014), 519528. 
[2] 
Ph. Baptiste, L. Peridy, E. Pinson, A branch and bound to minimize the number of late jobs on a single machine with release time constraints, European Journal of Operational Research, 144 (2003), 111. 
[3] 
J. C. Bean, Genetics and random keys for sequencing and optimization, ORSA Journal on Computing, 6 (1994), 154160. 
[4] 
R. L. Bulfin, R. M'hallah, Minimizing the weighted number of tardy jobs on a twomachine flow shop, Computers & Operations Research, 30 (2003), 18871900. doi: 10.1016/S03050548(02)001144. 
[5] 
W. Chen, Minimizing number of tardy jobs on a single machine subject to periodic maintenance, Omega, 37 (2009), 591599. 
[6] 
T. Chiang, L. Fu, A rulecentric memetic algorithm to minimize the number of tardy jobs in the job shop, International Journal of Production Research, 46 (2008), 69136931. 
[7] 
S. W. Choi, Y. D. Kim, Minimizing total tardiness on a twomachine reentrant flowshop, European Journal of Operational Research, 199 (2009), 375384. doi: 10.1016/j.ejor.2008.11.037. 
[8] 
H. S. Choi, D. H. Lee, Scheduling algorithms to minimize the number of tardy jobs in twostage hybrid flow shops, Computers & Industrial Engineering, 56 (2009), 113120. 
[9] 
D. T. Connolly, An improved annealing scheme for the QAP, European Journal of Operational Research, 46 (1990), 93100. doi: 10.1016/03772217(90)90301Q. 
[10] 
C. Desprez, F. Chu, C. Chu, Minimising the weighted number of tardy jobs in a hybrid flow shop with genetic algorithm, International Journal of Computer Integrated Manufacturing, 22 (2009), 745757. 
[11] 
X. Feng, F. Zheng, Y. Xu, Robust scheduling of a twostage hybrid flow shop with uncertain interval processing times, International Journal of Production Research, 54 (2016), 37063717. 
[12] 
E. M. GonzálezNeira, J. R. MontoyaTorres, D. Barrera, Flowshop scheduling problem under uncertainties: Review and trends, International Journal of Industrial Engineering Computations, 8 (2017), 399426. 
[13] 
A. M. A. Hariri, C. N. Potts, A branch and bound algorithm to minimize the number of late jobs in a permutation flowshop, European Journal of Operational Research, 38 (1989), 228237. doi: 10.1016/03772217(89)901070. 
[14] 
Ch. He, Y. Lin, J. Yuan, A note on the single machine scheduling to minimize the number of tardy jobs with deadlines, European Journal of Operational Research, 201 (2010), 966970. doi: 10.1016/j.ejor.2009.05.013. 
[15] 
F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan, M. Yazadn Parast, Genetic algorithm for bicriteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics and Computation, 194 (2007), 552560. doi: 10.1016/j.amc.2007.04.063. 
[16] 
J. Lenstra, A. Rinnooy Kan, P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343362. 
[17] 
G. Sh. Liu, Y. Zhou, H. D. Yang, Minimizing energy consumption and tardiness penalty for fuzzy flow shop scheduling with statedependent setup time, Journal of Cleaner Production, 147 (2017), 470484. 
[18] 
M. Liu, Sh. Wang, Ch. Chu, F. Chu, An improved exact algorithm for singlemachine scheduling to minimise the number of tardy jobs with periodic maintenance, International Journal of Production Research, 54 (2016), 35913602. 
[19] 
E. Lodree, W. Jang, C. M. Klein, A new rule for minimizing the number of tardy jobs in dynamic flow shops, European Journal of Operational Research, 159 (2004), 258263. doi: 10.1016/S03772217(03)004041. 
[20] 
M. Lundy, A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111124. doi: 10.1007/BF01582166. 
[21] 
R. M'Hallah, T. AlKhamis, A Benders decomposition approach to the weighted number of tardy jobs scheduling problem on unrelated parallel machines with production costs, International Journal of Production Research, 53 (2015), 59775987. 
[22] 
R. M'Hallah, R. L. Bulfin, Minimizing the weighted number of tardy jobs on a single machine with release dates, European Journal of Operational Research, 176 (2007), 727744. doi: 10.1016/j.ejor.2005.08.013. 
[23] 
M. Mirabi, SMT. Fatemi Ghomi, F. Jolai, A novel hybrid genetic algorithm to solve the maketoorder sequencedependent flowshop scheduling problem, Journal of Industrial Engineering International, 10 (2014), 19. 
[24] 
E. Molaee, Gh. Moslehi, M. Reisi, Minimizing maximum earliness and number of tardy jobs in the single machine scheduling problem, Computers & Mathematics with Applications, 60 (2010), 29092919. doi: 10.1016/j.camwa.2010.09.046. 
[25] 
J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Science, 15 (1968), 102109. 
[26] 
G. Mosheiov, A. Sarig, Minimum weighted number of tardy jobs on an mmachine flowshop with a critical machine, European Journal of Operational Research, 201 (2010), 404408. doi: 10.1016/j.ejor.2009.03.018. 
[27] 
Gh. Moslehi, M. Abouei Ardakan, Minimizing the number of Tardy jobs in a twomachine flowshop problem with nonsimultaneous job entrance, International Journal of Industrial Engineering & amp; Production Management, 23 (2013), 389400. 
[28] 
Gh. Moslehi, A. Jafari, Minimizing the number of tardy jobs under piecewiselinear deterioration, Computers & Industrial Engineering, 59 (2010), 573584. 
[29] 
B. Naderi, E. Najafi, M. Yazdani, An electromagnetismlike metaheuristic for openshop problems with no buffer, Journal of Industrial Engineering International, 8 (2012), 18. 
[30] 
S. NooriDarvish, R. TavakkoliMoghaddam, Minimizing the total tardiness and makespan in an open shop scheduling problem with sequencedependent setup times, Journal of Industrial Engineering International, 8 (2012), 113. 
[31] 
M. Rohaninejad, A. Saman Kheirkhah, B. Vahedi Nouri, P. Fattahi, Two hybrid tabu searchfirefly algorithms for the capacitated job shop scheduling problem with sequencedependent setup cost, International Journal of Computer Integrated Manufacturing, 28 (2015), 470487. 
[32] 
A. J. RuizTorres, G. Centeno, Minimizing the number of late jobs for the permutation flowshop problem with secondary resources, Computers & Operations Research, 35 (2008), 12271249. doi: 10.1016/j.cor.2006.07.013. 
[33] 
A. J. RuizTorres, J. H. AblanedoRosas, J. C. Ho, Minimizing the number of tardy jobs in the flowshop problem with operation and resource flexibility, Computers & Operations Research, 37 (2010), 282291. doi: 10.1016/j.cor.2009.04.018. 
[34] 
R. Şahin, O. Türkbey, A simulated annealing algorithm to find approximate Pareto optimal solutions for the multiobjective facility layout problem, The International Journal of Advanced Manufacturing Technology, 41 (2009), 10031018. 
[35] 
R. Sadykov, A branchandcheck algorithm for minimizing the weighted number of late jobs on a single machine with release dates, European Journal of Operational Research, 189 (2008), 12841304. doi: 10.1016/j.ejor.2006.06.078. 
[36] 
G. Wan, B. PC. Yen, Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs, European Journal of Operational Research, 195 (2009), 8997. doi: 10.1016/j.ejor.2008.01.029. 
[37] 
Zh. Wang, C. M. Wei, L. Sun, Solution algorithms for the number of tardy jobs minimisation scheduling with a timedependent learning effect, International Journal of Production Research, (2016), 18. 
[38] 
J. Yuan, Multiagent scheduling on a single machine with a fixed number of competing agents to minimize the weighted sum of number of tardy jobs and makespans, Journal of Combinatorial Optimization, 34 (2017), 443440. doi: 10.1007/s1087801600789. 
show all references
References:
[1] 
M. Abouei Ardakan, A. Hakimian, M. T. Rezvan, A branchandbound algorithm for minimising the number of tardy jobs in a twomachine flowshop problem with release datest, International Journal of Computer Integrated Manufacturing, 27 (2014), 519528. 
[2] 
Ph. Baptiste, L. Peridy, E. Pinson, A branch and bound to minimize the number of late jobs on a single machine with release time constraints, European Journal of Operational Research, 144 (2003), 111. 
[3] 
J. C. Bean, Genetics and random keys for sequencing and optimization, ORSA Journal on Computing, 6 (1994), 154160. 
[4] 
R. L. Bulfin, R. M'hallah, Minimizing the weighted number of tardy jobs on a twomachine flow shop, Computers & Operations Research, 30 (2003), 18871900. doi: 10.1016/S03050548(02)001144. 
[5] 
W. Chen, Minimizing number of tardy jobs on a single machine subject to periodic maintenance, Omega, 37 (2009), 591599. 
[6] 
T. Chiang, L. Fu, A rulecentric memetic algorithm to minimize the number of tardy jobs in the job shop, International Journal of Production Research, 46 (2008), 69136931. 
[7] 
S. W. Choi, Y. D. Kim, Minimizing total tardiness on a twomachine reentrant flowshop, European Journal of Operational Research, 199 (2009), 375384. doi: 10.1016/j.ejor.2008.11.037. 
[8] 
H. S. Choi, D. H. Lee, Scheduling algorithms to minimize the number of tardy jobs in twostage hybrid flow shops, Computers & Industrial Engineering, 56 (2009), 113120. 
[9] 
D. T. Connolly, An improved annealing scheme for the QAP, European Journal of Operational Research, 46 (1990), 93100. doi: 10.1016/03772217(90)90301Q. 
[10] 
C. Desprez, F. Chu, C. Chu, Minimising the weighted number of tardy jobs in a hybrid flow shop with genetic algorithm, International Journal of Computer Integrated Manufacturing, 22 (2009), 745757. 
[11] 
X. Feng, F. Zheng, Y. Xu, Robust scheduling of a twostage hybrid flow shop with uncertain interval processing times, International Journal of Production Research, 54 (2016), 37063717. 
[12] 
E. M. GonzálezNeira, J. R. MontoyaTorres, D. Barrera, Flowshop scheduling problem under uncertainties: Review and trends, International Journal of Industrial Engineering Computations, 8 (2017), 399426. 
[13] 
A. M. A. Hariri, C. N. Potts, A branch and bound algorithm to minimize the number of late jobs in a permutation flowshop, European Journal of Operational Research, 38 (1989), 228237. doi: 10.1016/03772217(89)901070. 
[14] 
Ch. He, Y. Lin, J. Yuan, A note on the single machine scheduling to minimize the number of tardy jobs with deadlines, European Journal of Operational Research, 201 (2010), 966970. doi: 10.1016/j.ejor.2009.05.013. 
[15] 
F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan, M. Yazadn Parast, Genetic algorithm for bicriteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics and Computation, 194 (2007), 552560. doi: 10.1016/j.amc.2007.04.063. 
[16] 
J. Lenstra, A. Rinnooy Kan, P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343362. 
[17] 
G. Sh. Liu, Y. Zhou, H. D. Yang, Minimizing energy consumption and tardiness penalty for fuzzy flow shop scheduling with statedependent setup time, Journal of Cleaner Production, 147 (2017), 470484. 
[18] 
M. Liu, Sh. Wang, Ch. Chu, F. Chu, An improved exact algorithm for singlemachine scheduling to minimise the number of tardy jobs with periodic maintenance, International Journal of Production Research, 54 (2016), 35913602. 
[19] 
E. Lodree, W. Jang, C. M. Klein, A new rule for minimizing the number of tardy jobs in dynamic flow shops, European Journal of Operational Research, 159 (2004), 258263. doi: 10.1016/S03772217(03)004041. 
[20] 
M. Lundy, A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111124. doi: 10.1007/BF01582166. 
[21] 
R. M'Hallah, T. AlKhamis, A Benders decomposition approach to the weighted number of tardy jobs scheduling problem on unrelated parallel machines with production costs, International Journal of Production Research, 53 (2015), 59775987. 
[22] 
R. M'Hallah, R. L. Bulfin, Minimizing the weighted number of tardy jobs on a single machine with release dates, European Journal of Operational Research, 176 (2007), 727744. doi: 10.1016/j.ejor.2005.08.013. 
[23] 
M. Mirabi, SMT. Fatemi Ghomi, F. Jolai, A novel hybrid genetic algorithm to solve the maketoorder sequencedependent flowshop scheduling problem, Journal of Industrial Engineering International, 10 (2014), 19. 
[24] 
E. Molaee, Gh. Moslehi, M. Reisi, Minimizing maximum earliness and number of tardy jobs in the single machine scheduling problem, Computers & Mathematics with Applications, 60 (2010), 29092919. doi: 10.1016/j.camwa.2010.09.046. 
[25] 
J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Science, 15 (1968), 102109. 
[26] 
G. Mosheiov, A. Sarig, Minimum weighted number of tardy jobs on an mmachine flowshop with a critical machine, European Journal of Operational Research, 201 (2010), 404408. doi: 10.1016/j.ejor.2009.03.018. 
[27] 
Gh. Moslehi, M. Abouei Ardakan, Minimizing the number of Tardy jobs in a twomachine flowshop problem with nonsimultaneous job entrance, International Journal of Industrial Engineering & amp; Production Management, 23 (2013), 389400. 
[28] 
Gh. Moslehi, A. Jafari, Minimizing the number of tardy jobs under piecewiselinear deterioration, Computers & Industrial Engineering, 59 (2010), 573584. 
[29] 
B. Naderi, E. Najafi, M. Yazdani, An electromagnetismlike metaheuristic for openshop problems with no buffer, Journal of Industrial Engineering International, 8 (2012), 18. 
[30] 
S. NooriDarvish, R. TavakkoliMoghaddam, Minimizing the total tardiness and makespan in an open shop scheduling problem with sequencedependent setup times, Journal of Industrial Engineering International, 8 (2012), 113. 
[31] 
M. Rohaninejad, A. Saman Kheirkhah, B. Vahedi Nouri, P. Fattahi, Two hybrid tabu searchfirefly algorithms for the capacitated job shop scheduling problem with sequencedependent setup cost, International Journal of Computer Integrated Manufacturing, 28 (2015), 470487. 
[32] 
A. J. RuizTorres, G. Centeno, Minimizing the number of late jobs for the permutation flowshop problem with secondary resources, Computers & Operations Research, 35 (2008), 12271249. doi: 10.1016/j.cor.2006.07.013. 
[33] 
A. J. RuizTorres, J. H. AblanedoRosas, J. C. Ho, Minimizing the number of tardy jobs in the flowshop problem with operation and resource flexibility, Computers & Operations Research, 37 (2010), 282291. doi: 10.1016/j.cor.2009.04.018. 
[34] 
R. Şahin, O. Türkbey, A simulated annealing algorithm to find approximate Pareto optimal solutions for the multiobjective facility layout problem, The International Journal of Advanced Manufacturing Technology, 41 (2009), 10031018. 
[35] 
R. Sadykov, A branchandcheck algorithm for minimizing the weighted number of late jobs on a single machine with release dates, European Journal of Operational Research, 189 (2008), 12841304. doi: 10.1016/j.ejor.2006.06.078. 
[36] 
G. Wan, B. PC. Yen, Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs, European Journal of Operational Research, 195 (2009), 8997. doi: 10.1016/j.ejor.2008.01.029. 
[37] 
Zh. Wang, C. M. Wei, L. Sun, Solution algorithms for the number of tardy jobs minimisation scheduling with a timedependent learning effect, International Journal of Production Research, (2016), 18. 
[38] 
J. Yuan, Multiagent scheduling on a single machine with a fixed number of competing agents to minimize the weighted sum of number of tardy jobs and makespans, Journal of Combinatorial Optimization, 34 (2017), 443440. doi: 10.1007/s1087801600789. 
Influential Parameter  High setting  Low setting 
 Jobs are available in a longer time interval after the beginning of scheduling  Jobs are available in a short time interval which is closer to the close to the beginning of scheduling 
 Tight due date intervals  Loose due date intervals 
 Different due dates  Similar due dates 
Number of tardy jobs  More tardy jobs  Less tardy jobs 
Influential Parameter  High setting  Low setting 
 Jobs are available in a longer time interval after the beginning of scheduling  Jobs are available in a short time interval which is closer to the close to the beginning of scheduling 
 Tight due date intervals  Loose due date intervals 
 Different due dates  Similar due dates 
Number of tardy jobs  More tardy jobs  Less tardy jobs 
CPU time (Sec.) for PSAGA  Number of problems optimally solved^{*}    
n  min  avg  max  OPT#  OPT(%)  avg  max  avg  max  
5  0.00  0.09  0.17  20  100%  1.00  1.00  1.06  1.50  
6  0.05  0.16  0.28  20  100%  1.00  1.00  1.11  1.33  
7  0.02  0.20  0.37  20  100%  1.00  1.00  1.09  1.40  
8  0.02  0.18  0.37  20  100%  1.00  1.00  1.10  1.20  
9  0.02  0.21  0.45  20  100%  1.00  1.00  1.06  1.17  
10  0.08  0.16  0.41  20  100%  1.00  1.00  1.12  1.29  
12  0.02  0.06  0.50  20   1.00  1.00  1.01  1.10  
14  0.23  0.41  0.86  20   1.00  1.00  1.12  1.25  
16  0.33  0.70  1.15  19   1.00  1.07  1.11  1.25  
18  0.62  1.34  2.11  19  1.01  1.07  1.04  1.13  
20  0.44  0.90  1.67  19   1.01  1.08  1.05  1.12 
CPU time (Sec.) for PSAGA  Number of problems optimally solved^{*}    
n  min  avg  max  OPT#  OPT(%)  avg  max  avg  max  
5  0.00  0.09  0.17  20  100%  1.00  1.00  1.06  1.50  
6  0.05  0.16  0.28  20  100%  1.00  1.00  1.11  1.33  
7  0.02  0.20  0.37  20  100%  1.00  1.00  1.09  1.40  
8  0.02  0.18  0.37  20  100%  1.00  1.00  1.10  1.20  
9  0.02  0.21  0.45  20  100%  1.00  1.00  1.06  1.17  
10  0.08  0.16  0.41  20  100%  1.00  1.00  1.12  1.29  
12  0.02  0.06  0.50  20   1.00  1.00  1.01  1.10  
14  0.23  0.41  0.86  20   1.00  1.00  1.12  1.25  
16  0.33  0.70  1.15  19   1.00  1.07  1.11  1.25  
18  0.62  1.34  2.11  19  1.01  1.07  1.04  1.13  
20  0.44  0.90  1.67  19   1.01  1.08  1.05  1.12 
CPU time (Sec.) for PSAGA  Number of problems optimally solved^{*}    
n  min  avg  max  OPT#  OPT(%)  avg  max  avg  max  
5  0.02  0.09  0.16  20  100%  1.00  1.00  1.10  1.33  
6  0.06  0.15  0.31  20  100%  1.00  1.00  1.10  1.25  
7  0.14  0.27  0.38  20  100%  1.00  1.00  1.05  1.20  
8  0.11  0.20  0.34  20  100%  1.00  1.00  1.07  1.40  
9  0.23  0.41  0.55  20  100%  1.00  1.00  1.06  1.33  
10  0.11  0.29  0.55  20  100%  1.00  1.00  1.09  1.60  
12  0.22  0.33  0.61  20   1.00  1.00  1.07  1.43  
14  0.22  0.48  0.67  20   1.00  1.00  1.08  1.20  
16  0.37  0.70  1.08  20   1.00  1.00  1.09  1.17  
18  0.70  1.14  1.62  19  1.01  1.03  1.06  1.15  
20  0.78  1.12  2.29  19   1.01  1.03  1.07  1.12 
CPU time (Sec.) for PSAGA  Number of problems optimally solved^{*}    
n  min  avg  max  OPT#  OPT(%)  avg  max  avg  max  
5  0.02  0.09  0.16  20  100%  1.00  1.00  1.10  1.33  
6  0.06  0.15  0.31  20  100%  1.00  1.00  1.10  1.25  
7  0.14  0.27  0.38  20  100%  1.00  1.00  1.05  1.20  
8  0.11  0.20  0.34  20  100%  1.00  1.00  1.07  1.40  
9  0.23  0.41  0.55  20  100%  1.00  1.00  1.06  1.33  
10  0.11  0.29  0.55  20  100%  1.00  1.00  1.09  1.60  
12  0.22  0.33  0.61  20   1.00  1.00  1.07  1.43  
14  0.22  0.48  0.67  20   1.00  1.00  1.08  1.20  
16  0.37  0.70  1.08  20   1.00  1.00  1.09  1.17  
18  0.70  1.14  1.62  19  1.01  1.03  1.06  1.15  
20  0.78  1.12  2.29  19   1.01  1.03  1.07  1.12 
CPU time (Sec.) for PSAGA  CPU times(Sec.) for HEU1   
n  min  avg  max  min  avg  max  avg  max  
30  1.06  2.16  3.45  0.00  0.00  0.01  1.16  1.26  
40  1.44  2.01  2.87  0.01  0.01  0.01  1.13  1.31  
50  2.79  4.90  6.72  0.01  0.02  0.03  1.11  1.20  
60  5.01  7.96  11.06  0.00  0.01  0.04  1.11  1.21  
70  6.44  10.63  14.45  0.01  0.02  0.04  1.11  1.25  
80  11.29  18.65  26.46  0.03  0.04  0.06  1.08  1.22  
90  10.42  23.23  36.60  0.02  0.04  0.07  1.07  1.14  
100  13.29  19.39  24.73  0.03  0.05  0.09  1.06  1.10 
CPU time (Sec.) for PSAGA  CPU times(Sec.) for HEU1   
n  min  avg  max  min  avg  max  avg  max  
30  1.06  2.16  3.45  0.00  0.00  0.01  1.16  1.26  
40  1.44  2.01  2.87  0.01  0.01  0.01  1.13  1.31  
50  2.79  4.90  6.72  0.01  0.02  0.03  1.11  1.20  
60  5.01  7.96  11.06  0.00  0.01  0.04  1.11  1.21  
70  6.44  10.63  14.45  0.01  0.02  0.04  1.11  1.25  
80  11.29  18.65  26.46  0.03  0.04  0.06  1.08  1.22  
90  10.42  23.23  36.60  0.02  0.04  0.07  1.07  1.14  
100  13.29  19.39  24.73  0.03  0.05  0.09  1.06  1.10 
CPU time (Sec.) for PSAGA  CPU times(Sec.) for HEU1   
n  min  avg  max  min  avg  max  avg  max  
30  1.53  2.52  3.89  0.00  0.00  0.01  1.06  1.13  
40  2.78  4.16  6.07  0.01  0.01  0.01  1.05  1.10  
50  4.54  6.83  9.19  0.01  0.01  0.03  1.05  1.10  
60  6.25  9.65  12.78  0.00  0.02  0.03  1.03  1.06  
70  8.42  12.07  19.63  0.02  0.03  0.04  1.03  1.09  
80  13.34  17.40  28.78  0.01  0.04  0.08  1.04  1.08  
90  15.34  22.24  28.78  0.03  0.06  0.09  1.03  1.07  
100  17.82  38.95  58.25  0.05  0.10  0.14  1.03  1.07 
CPU time (Sec.) for PSAGA  CPU times(Sec.) for HEU1   
n  min  avg  max  min  avg  max  avg  max  
30  1.53  2.52  3.89  0.00  0.00  0.01  1.06  1.13  
40  2.78  4.16  6.07  0.01  0.01  0.01  1.05  1.10  
50  4.54  6.83  9.19  0.01  0.01  0.03  1.05  1.10  
60  6.25  9.65  12.78  0.00  0.02  0.03  1.03  1.06  
70  8.42  12.07  19.63  0.02  0.03  0.04  1.03  1.09  
80  13.34  17.40  28.78  0.01  0.04  0.08  1.04  1.08  
90  15.34  22.24  28.78  0.03  0.06  0.09  1.03  1.07  
100  17.82  38.95  58.25  0.05  0.10  0.14  1.03  1.07 
CPU time (Sec.) set High  CPU times(Sec.) set Low  
n  min  avg  max  min  avg  max  
200  0.08  0.13  0.17  0.22  0.25  0.30  
400  0.70  1.05  1.33  1.64  1.85  2.06  
600  2.31  3.03  3.64  5.60  6.13  6.55  
60  6.25  9.65  12.78  0.00  0.02  0.03  
800  5.53  7.10  8.64  12.97  14.46  19.42  
1000  11.16  13.98  16.37  25.55  32.08  43.46 
CPU time (Sec.) set High  CPU times(Sec.) set Low  
n  min  avg  max  min  avg  max  
200  0.08  0.13  0.17  0.22  0.25  0.30  
400  0.70  1.05  1.33  1.64  1.85  2.06  
600  2.31  3.03  3.64  5.60  6.13  6.55  
60  6.25  9.65  12.78  0.00  0.02  0.03  
800  5.53  7.10  8.64  12.97  14.46  19.42  
1000  11.16  13.98  16.37  25.55  32.08  43.46 
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