# American Institute of Mathematical Sciences

December 2017, 7(4): 465-480. doi: 10.3934/naco.2017029

## A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine 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, 84156-83111, Isfahan, Iran

* Corresponding author

Received  December 2016 Revised  September 2017 Published  October 2017

Fund Project: The reviewing process of this paper was handled by Associate Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey, at the occasion of 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016

In this paper, first, the problem of dynamic two-machine 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 branch-and-bound algorithm which has been already developed in the literature. Computational experiments demonstrate that the proposed hybrid algorithm can solve the entire small-sized problems and more than 95% of medium-sized problems optimally.

Citation: Mostafa Abouei Ardakan, A. Kourank Beheshti, S. Hamid Mirmohammadi, Hamed Davari Ardakani. A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 465-480. doi: 10.3934/naco.2017029
##### References:
 [1] M. Abouei Ardakan, A. Hakimian and M. T. Rezvan, A branch-and-bound algorithm for minimising the number of tardy jobs in a two-machine flow-shop problem with release datest, International Journal of Computer Integrated Manufacturing, 27 (2014), 519-528. [2] Ph. Baptiste, L. Peridy and 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), 1-11. [3] J. C. Bean, Genetics and random keys for sequencing and optimization, ORSA Journal on Computing, 6 (1994), 154-160. [4] R. L. Bulfin and R. M'hallah, Minimizing the weighted number of tardy jobs on a two-machine flow shop, Computers & Operations Research, 30 (2003), 1887-1900. doi: 10.1016/S0305-0548(02)00114-4. [5] W. Chen, Minimizing number of tardy jobs on a single machine subject to periodic maintenance, Omega, 37 (2009), 591-599. [6] T. Chiang and L. Fu, A rule-centric memetic algorithm to minimize the number of tardy jobs in the job shop, International Journal of Production Research, 46 (2008), 6913-6931. [7] S. W. Choi and Y. D. Kim, Minimizing total tardiness on a two-machine re-entrant flowshop, European Journal of Operational Research, 199 (2009), 375-384. doi: 10.1016/j.ejor.2008.11.037. [8] H. S. Choi and D. H. Lee, Scheduling algorithms to minimize the number of tardy jobs in two-stage hybrid flow shops, Computers & Industrial Engineering, 56 (2009), 113-120. [9] D. T. Connolly, An improved annealing scheme for the QAP, European Journal of Operational Research, 46 (1990), 93-100. doi: 10.1016/0377-2217(90)90301-Q. [10] C. Desprez, F. Chu and 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), 745-757. [11] X. Feng, F. Zheng and Y. Xu, Robust scheduling of a two-stage hybrid flow shop with uncertain interval processing times, International Journal of Production Research, 54 (2016), 3706-3717. [12] E. M. González-Neira, J. R. Montoya-Torres and D. Barrera, Flow-shop scheduling problem under uncertainties: Review and trends, International Journal of Industrial Engineering Computations, 8 (2017), 399-426. [13] A. M. A. Hariri and C. N. Potts, A branch and bound algorithm to minimize the number of late jobs in a permutation flow-shop, European Journal of Operational Research, 38 (1989), 228-237. doi: 10.1016/0377-2217(89)90107-0. [14] Ch. He, Y. Lin and 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), 966-970. doi: 10.1016/j.ejor.2009.05.013. [15] F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan and M. Yazadn Parast, Genetic algorithm for bi-criteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics and Computation, 194 (2007), 552-560. doi: 10.1016/j.amc.2007.04.063. [16] J. Lenstra, A. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343-362. [17] G. Sh. Liu, Y. Zhou and H. D. Yang, Minimizing energy consumption and tardiness penalty for fuzzy flow shop scheduling with state-dependent setup time, Journal of Cleaner Production, 147 (2017), 470-484. [18] M. Liu, Sh. Wang, Ch. Chu and F. Chu, An improved exact algorithm for single-machine scheduling to minimise the number of tardy jobs with periodic maintenance, International Journal of Production Research, 54 (2016), 3591-3602. [19] E. Lodree, W. Jang and C. M. Klein, A new rule for minimizing the number of tardy jobs in dynamic flow shops, European Journal of Operational Research, 159 (2004), 258-263. doi: 10.1016/S0377-2217(03)00404-1. [20] M. Lundy and A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111-124. doi: 10.1007/BF01582166. [21] R. M'Hallah and T. Al-Khamis, 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), 5977-5987. [22] R. M'Hallah and R. L. Bulfin, Minimizing the weighted number of tardy jobs on a single machine with release dates, European Journal of Operational Research, 176 (2007), 727-744. doi: 10.1016/j.ejor.2005.08.013. [23] M. Mirabi, SMT. Fatemi Ghomi and F. Jolai, A novel hybrid genetic algorithm to solve the make-to-order sequence-dependent flow-shop scheduling problem, Journal of Industrial Engineering International, 10 (2014), 1-9. [24] E. Molaee, Gh. Moslehi and M. Reisi, Minimizing maximum earliness and number of tardy jobs in the single machine scheduling problem, Computers & Mathematics with Applications, 60 (2010), 2909-2919. 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), 102-109. [26] G. Mosheiov and A. Sarig, Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine, European Journal of Operational Research, 201 (2010), 404-408. doi: 10.1016/j.ejor.2009.03.018. [27] Gh. Moslehi and M. Abouei Ardakan, Minimizing the number of Tardy jobs in a two-machine flowshop problem with non-simultaneous job entrance, International Journal of Industrial Engineering & amp; Production Management, 23 (2013), 389-400. [28] Gh. Moslehi and A. Jafari, Minimizing the number of tardy jobs under piecewise-linear deterioration, Computers & Industrial Engineering, 59 (2010), 573-584. [29] B. Naderi, E. Najafi and M. Yazdani, An electromagnetism-like metaheuristic for open-shop problems with no buffer, Journal of Industrial Engineering International, 8 (2012), 1-8. [30] S. Noori-Darvish and R. Tavakkoli-Moghaddam, Minimizing the total tardiness and makespan in an open shop scheduling problem with sequence-dependent setup times, Journal of Industrial Engineering International, 8 (2012), 1-13. [31] M. Rohaninejad, A. Saman Kheirkhah, B. Vahedi Nouri and P. Fattahi, Two hybrid tabu search-firefly algorithms for the capacitated job shop scheduling problem with sequence-dependent setup cost, International Journal of Computer Integrated Manufacturing, 28 (2015), 470-487. [32] A. J. Ruiz-Torres and G. Centeno, Minimizing the number of late jobs for the permutation flowshop problem with secondary resources, Computers & Operations Research, 35 (2008), 1227-1249. doi: 10.1016/j.cor.2006.07.013. [33] A. J. Ruiz-Torres, J. H. Ablanedo-Rosas and J. C. Ho, Minimizing the number of tardy jobs in the flowshop problem with operation and resource flexibility, Computers & Operations Research, 37 (2010), 282-291. doi: 10.1016/j.cor.2009.04.018. [34] R. Şahin and O. Türkbey, A simulated annealing algorithm to find approximate Pareto optimal solutions for the multi-objective facility layout problem, The International Journal of Advanced Manufacturing Technology, 41 (2009), 1003-1018. [35] R. Sadykov, A branch-and-check algorithm for minimizing the weighted number of late jobs on a single machine with release dates, European Journal of Operational Research, 189 (2008), 1284-1304. doi: 10.1016/j.ejor.2006.06.078. [36] G. Wan and B. P-C. Yen, Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs, European Journal of Operational Research, 195 (2009), 89-97. doi: 10.1016/j.ejor.2008.01.029. [37] Zh. Wang, C. M. Wei and L. Sun, Solution algorithms for the number of tardy jobs minimisation scheduling with a time-dependent learning effect, International Journal of Production Research, (2016), 1-8. [38] J. Yuan, Multi-agent 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), 443-440. doi: 10.1007/s10878-016-0078-9.

show all references

##### References:
 [1] M. Abouei Ardakan, A. Hakimian and M. T. Rezvan, A branch-and-bound algorithm for minimising the number of tardy jobs in a two-machine flow-shop problem with release datest, International Journal of Computer Integrated Manufacturing, 27 (2014), 519-528. [2] Ph. Baptiste, L. Peridy and 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), 1-11. [3] J. C. Bean, Genetics and random keys for sequencing and optimization, ORSA Journal on Computing, 6 (1994), 154-160. [4] R. L. Bulfin and R. M'hallah, Minimizing the weighted number of tardy jobs on a two-machine flow shop, Computers & Operations Research, 30 (2003), 1887-1900. doi: 10.1016/S0305-0548(02)00114-4. [5] W. Chen, Minimizing number of tardy jobs on a single machine subject to periodic maintenance, Omega, 37 (2009), 591-599. [6] T. Chiang and L. Fu, A rule-centric memetic algorithm to minimize the number of tardy jobs in the job shop, International Journal of Production Research, 46 (2008), 6913-6931. [7] S. W. Choi and Y. D. Kim, Minimizing total tardiness on a two-machine re-entrant flowshop, European Journal of Operational Research, 199 (2009), 375-384. doi: 10.1016/j.ejor.2008.11.037. [8] H. S. Choi and D. H. Lee, Scheduling algorithms to minimize the number of tardy jobs in two-stage hybrid flow shops, Computers & Industrial Engineering, 56 (2009), 113-120. [9] D. T. Connolly, An improved annealing scheme for the QAP, European Journal of Operational Research, 46 (1990), 93-100. doi: 10.1016/0377-2217(90)90301-Q. [10] C. Desprez, F. Chu and 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), 745-757. [11] X. Feng, F. Zheng and Y. Xu, Robust scheduling of a two-stage hybrid flow shop with uncertain interval processing times, International Journal of Production Research, 54 (2016), 3706-3717. [12] E. M. González-Neira, J. R. Montoya-Torres and D. Barrera, Flow-shop scheduling problem under uncertainties: Review and trends, International Journal of Industrial Engineering Computations, 8 (2017), 399-426. [13] A. M. A. Hariri and C. N. Potts, A branch and bound algorithm to minimize the number of late jobs in a permutation flow-shop, European Journal of Operational Research, 38 (1989), 228-237. doi: 10.1016/0377-2217(89)90107-0. [14] Ch. He, Y. Lin and 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), 966-970. doi: 10.1016/j.ejor.2009.05.013. [15] F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan and M. Yazadn Parast, Genetic algorithm for bi-criteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics and Computation, 194 (2007), 552-560. doi: 10.1016/j.amc.2007.04.063. [16] J. Lenstra, A. Rinnooy Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343-362. [17] G. Sh. Liu, Y. Zhou and H. D. Yang, Minimizing energy consumption and tardiness penalty for fuzzy flow shop scheduling with state-dependent setup time, Journal of Cleaner Production, 147 (2017), 470-484. [18] M. Liu, Sh. Wang, Ch. Chu and F. Chu, An improved exact algorithm for single-machine scheduling to minimise the number of tardy jobs with periodic maintenance, International Journal of Production Research, 54 (2016), 3591-3602. [19] E. Lodree, W. Jang and C. M. Klein, A new rule for minimizing the number of tardy jobs in dynamic flow shops, European Journal of Operational Research, 159 (2004), 258-263. doi: 10.1016/S0377-2217(03)00404-1. [20] M. Lundy and A. Mees, Convergence of an annealing algorithm, Mathematical Programming, 34 (1986), 111-124. doi: 10.1007/BF01582166. [21] R. M'Hallah and T. Al-Khamis, 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), 5977-5987. [22] R. M'Hallah and R. L. Bulfin, Minimizing the weighted number of tardy jobs on a single machine with release dates, European Journal of Operational Research, 176 (2007), 727-744. doi: 10.1016/j.ejor.2005.08.013. [23] M. Mirabi, SMT. Fatemi Ghomi and F. Jolai, A novel hybrid genetic algorithm to solve the make-to-order sequence-dependent flow-shop scheduling problem, Journal of Industrial Engineering International, 10 (2014), 1-9. [24] E. Molaee, Gh. Moslehi and M. Reisi, Minimizing maximum earliness and number of tardy jobs in the single machine scheduling problem, Computers & Mathematics with Applications, 60 (2010), 2909-2919. 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), 102-109. [26] G. Mosheiov and A. Sarig, Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine, European Journal of Operational Research, 201 (2010), 404-408. doi: 10.1016/j.ejor.2009.03.018. [27] Gh. Moslehi and M. Abouei Ardakan, Minimizing the number of Tardy jobs in a two-machine flowshop problem with non-simultaneous job entrance, International Journal of Industrial Engineering & amp; Production Management, 23 (2013), 389-400. [28] Gh. Moslehi and A. Jafari, Minimizing the number of tardy jobs under piecewise-linear deterioration, Computers & Industrial Engineering, 59 (2010), 573-584. [29] B. Naderi, E. Najafi and M. Yazdani, An electromagnetism-like metaheuristic for open-shop problems with no buffer, Journal of Industrial Engineering International, 8 (2012), 1-8. [30] S. Noori-Darvish and R. Tavakkoli-Moghaddam, Minimizing the total tardiness and makespan in an open shop scheduling problem with sequence-dependent setup times, Journal of Industrial Engineering International, 8 (2012), 1-13. [31] M. Rohaninejad, A. Saman Kheirkhah, B. Vahedi Nouri and P. Fattahi, Two hybrid tabu search-firefly algorithms for the capacitated job shop scheduling problem with sequence-dependent setup cost, International Journal of Computer Integrated Manufacturing, 28 (2015), 470-487. [32] A. J. Ruiz-Torres and G. Centeno, Minimizing the number of late jobs for the permutation flowshop problem with secondary resources, Computers & Operations Research, 35 (2008), 1227-1249. doi: 10.1016/j.cor.2006.07.013. [33] A. J. Ruiz-Torres, J. H. Ablanedo-Rosas and J. C. Ho, Minimizing the number of tardy jobs in the flowshop problem with operation and resource flexibility, Computers & Operations Research, 37 (2010), 282-291. doi: 10.1016/j.cor.2009.04.018. [34] R. Şahin and O. Türkbey, A simulated annealing algorithm to find approximate Pareto optimal solutions for the multi-objective facility layout problem, The International Journal of Advanced Manufacturing Technology, 41 (2009), 1003-1018. [35] R. Sadykov, A branch-and-check algorithm for minimizing the weighted number of late jobs on a single machine with release dates, European Journal of Operational Research, 189 (2008), 1284-1304. doi: 10.1016/j.ejor.2006.06.078. [36] G. Wan and B. P-C. Yen, Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs, European Journal of Operational Research, 195 (2009), 89-97. doi: 10.1016/j.ejor.2008.01.029. [37] Zh. Wang, C. M. Wei and L. Sun, Solution algorithms for the number of tardy jobs minimisation scheduling with a time-dependent learning effect, International Journal of Production Research, (2016), 1-8. [38] J. Yuan, Multi-agent 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), 443-440. doi: 10.1007/s10878-016-0078-9.
The flowchart of the proposed hybrid meta-heuristic algorithm
An example for R-K method
Crossover operator
The differences between two groups of generated problems
 Influential Parameter High setting Low setting $r_j$ 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 $D_j$ Tight due date intervals Loose due date intervals $D_j$ Different due dates Similar due dates Number of tardy jobs More tardy jobs Less tardy jobs
 Influential Parameter High setting Low setting $r_j$ 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 $D_j$ Tight due date intervals Loose due date intervals $D_j$ Different due dates Similar due dates Number of tardy jobs More tardy jobs Less tardy jobs
Performance of PSA-GA and HEU1 algorithms in set High (compared with B & B)
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 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 $100\%$ 1.00 1.00 1.01 1.10 14 0.23 0.41 0.86 20 $100\%$ 1.00 1.00 1.12 1.25 16 0.33 0.70 1.15 19 $95\%$ 1.00 1.07 1.11 1.25 18 0.62 1.34 2.11 19 $95\%$ 1.01 1.07 1.04 1.13 20 0.44 0.90 1.67 19 $95\%$ 1.01 1.08 1.05 1.12
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 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 $100\%$ 1.00 1.00 1.01 1.10 14 0.23 0.41 0.86 20 $100\%$ 1.00 1.00 1.12 1.25 16 0.33 0.70 1.15 19 $95\%$ 1.00 1.07 1.11 1.25 18 0.62 1.34 2.11 19 $95\%$ 1.01 1.07 1.04 1.13 20 0.44 0.90 1.67 19 $95\%$ 1.01 1.08 1.05 1.12
Performance of PSA-GA and HEU1 algorithms in set Low (compared with B & B)
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 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 $100\%$ 1.00 1.00 1.07 1.43 14 0.22 0.48 0.67 20 $100\%$ 1.00 1.00 1.08 1.20 16 0.37 0.70 1.08 20 $100\%$ 1.00 1.00 1.09 1.17 18 0.70 1.14 1.62 19 $95\%$ 1.01 1.03 1.06 1.15 20 0.78 1.12 2.29 19 $95\%$ 1.01 1.03 1.07 1.12
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 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 $100\%$ 1.00 1.00 1.07 1.43 14 0.22 0.48 0.67 20 $100\%$ 1.00 1.00 1.08 1.20 16 0.37 0.70 1.08 20 $100\%$ 1.00 1.00 1.09 1.17 18 0.70 1.14 1.62 19 $95\%$ 1.01 1.03 1.06 1.15 20 0.78 1.12 2.29 19 $95\%$ 1.01 1.03 1.07 1.12
Performance of algorithms HEU1 and PSA-GA in large problems; set High
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ 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 PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ 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
Performance of algorithms HEU1 and PSA-GA in large problems; set Low
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ 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 PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ 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
Results of the HEU1 algorithm; large-sized problem instances
 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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