doi: 10.3934/jimo.2019122

Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem

1. 

Department of Science and Technology Teaching, China University of Political Science and Law, Beijing 100088, China

2. 

College of Information Science and Engineering, Guangxi University for Nationalities, Key Laboratories of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China

3. 

School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

4. 

College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China

* Corresponding author: Yongquan Zhou

Received  July 2018 Revised  July 2019 Published  October 2019

The job-shop scheduling problem is one of the well-known hardest combinatorial optimization problems. The problem has captured the interest of a significant number of researchers, but no efficient solution algorithm has been found yet for solving it to optimality in polynomial time. In this paper, a hybrid social-spider optimization algorithm with differential mutation operator is presented to solve the job-shop scheduling problem. To improve the exploration capabilities of the social spider optimization algorithm (SSO), we incorporate the DM operator (a mutation operator taken from the deferential evolutionary (DE) algorithm) into the framework of the female cooperative operator. The experimental results show that the proposed method effectiveness in solving job-shop scheduling compared to other optimization algorithms in the literature.

Citation: Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019122
References:
[1]

R. F. Abdel-Kader, An improved PSO algorithm with genetic and neighborhood-based diversity operators for the job shop scheduling problem, Applied Artificial Intelligence, 32 (2018), 433-462.  doi: 10.1080/08839514.2018.1481903.  Google Scholar

[2]

M. Amirghasemi and R. Zamani, An effective asexual genetic algorithm for solving the job shop scheduling problem, Computers & Industrial Engineering, 83 (2015), 123-138.  doi: 10.1016/j.cie.2015.02.011.  Google Scholar

[3]

A. ElmiM. SolimanpurbS. Topaloglua and A. Elmic, A simulated annealing algorithm for the job shop cell scheduling problem with intercellular moves and reentrant parts, Computers & Industrial Engineering, 61 (2011), 171-178.  doi: 10.1016/j.cie.2011.03.007.  Google Scholar

[4]

J. E. Beasley, Or-library: Distributing test problems by electronic mail, J. of the Operational Research Society, 41 (1990), 1069-1072.  doi: 10.2307/2582903.  Google Scholar

[5]

E. Cuevas, M. A. Díaz Cortés and D. A. O. Navarro, Advances of Evolutionary Computation: Methods and Operators, Studies in Computational Intelligence, 629, Springer, 2016, 9–33. doi: 10.1007/978-3-319-28503-0.  Google Scholar

[6]

E. Cuevas, M. Cienfuegos, R. Rojas and A. Padilla, Computational Intelligence Applications in Modeling and Control, Studies in Computational Intelligence, 575, Springer, 2015, 123–146. doi: 10.1007/978-3-319-11017-2.  Google Scholar

[7]

E. CuevasM. CienfuegosD. Zaldivar and M. Perez-Cisneros, A swarm optimization algorithm inspired in the behavior of the social-spider, Expert Systems with Applications, 40 (2013), 6374-6384.  doi: 10.1016/j.eswa.2013.05.041.  Google Scholar

[8]

E. Cuevas, V. Osuna and D. Oliva, Evolutionary Computation Techniques: A Comparative Perspective, Studies in Computational Intelligence, 686 (2017), 65–93. doi: 10.1007/978-3-319-51109-2.  Google Scholar

[9]

T. K. DaoT. S. Pan and J. S. Pan, Parallel bat algorithm for optimizing makespan in job shop scheduling problems, J. of Intelligent Manufacturing, 29 (2018), 451-462.  doi: 10.1007/s10845-015-1121-x.  Google Scholar

[10]

N. FiǧlaliC. ÖzkaleO. Engin A. and Fi ǧlali, Investigation of Ant System parameter interactions by using design of experiments for job-shop scheduling problems, Computers & Industrial Engineering, 56 (2009), 538-559.  doi: 10.1016/j.cie.2007.06.001.  Google Scholar

[11]

H. Fisher and G. L. Thompson, Probabilistic learning combinations of local job-shop scheduling rules, in Industrial Scheduling, Prentice Hall, 1963, 225–251. Google Scholar

[12]

L. GaoX. LiX. WenC. Lu and F. Wen, A hybrid algorithm based on a new neighborhood structure evaluation method for job shop scheduling problem, Computers & Industrial Engineering, 88 (2015), 417-429.  doi: 10.1016/j.cie.2015.08.002.  Google Scholar

[13]

A. S. Jain and S. Meeran, Deterministic job-shop scheduling: Past, present and future, European J. of Operational Research, 113 (1999), 390-434.  doi: 10.1016/S0377-2217(98)00113-1.  Google Scholar

[14]

S. KavithaP. VenkumarN. Rajini and P. Pitchipoo, An efficient social spider optimization for flexible job shop scheduling problem, J. of Advanced Manufacturing Systems, 17 (2018), 181-196.  doi: 10.1142/S0219686718500117.  Google Scholar

[15]

M. Kurdi, A new hybrid island model genetic algorithm for job shop scheduling problem, Computers & Industrial Engineering, 88 (2015), 273-283.  doi: 10.1016/j.cie.2015.07.015.  Google Scholar

[16]

M. Kurdi, A Social Spider Optimization Algorithm for Hybrid Flow Shop Scheduling with Multiprocessor Task, 12th International NCM Conference: Challenges in Industrial Engineering & Operation Management, 2018. Google Scholar

[17]

M. Kurdi, An effective genetic algorithm with a critical-path-guided Giffler and Thompson crossover operator for job shop scheduling problem, International J. of Intelligent Systems and Applications in Engineering, 7 (2019), 13-18.  doi: 10.18201/ijisae.2019751247.  Google Scholar

[18]

M. Kurdi, An effective new island model genetic algorithm for job shop scheduling problem, Comput. Oper. Res., 67 (2016), 132-142.  doi: 10.1016/j.cor.2015.10.005.  Google Scholar

[19]

M. Kurdi, An improved island model memetic algorithm with a new cooperation phase for multi-objective job shop scheduling problem, Computers & Industrial Engineering, 111 (2017), 183-201.  doi: 10.1016/j.cie.2017.07.021.  Google Scholar

[20]

T.-L. LinS.-J. HorngT.-W. KaoY-.H. ChenR.-S. RunR.-J. ChenJ.-L. Lai and I.-H. Kuo, An efficient job-shop scheduling algorithm based on particle swarm optimization, Expert Systems with Applications, 37 (2010), 2629-2636.  doi: 10.1016/j.eswa.2009.08.015.  Google Scholar

[21]

M. LiuZ.-J. SunJ.-W. Yan and J.-S. Kang, An adaptive annealing genetic algorithm for the job-shop planning and scheduling problem, Expert Systems with Applications, 38 (2011), 9248-9255.  doi: 10.1016/j.eswa.2011.01.136.  Google Scholar

[22]

S. LuC. Sun and Z. Lu, An improved quantum-behaved particle swarm optimization method for short-term combined economic emission hydrothermal scheduling, Energy Conversion and Management, 51 (2010), 561-571.  doi: 10.1016/j.enconman.2009.10.024.  Google Scholar

[23]

T. B. Lubin, The Evolution of Sociality in Spiders, Advances in the Study of Behavior, 37 (2007), 83-145.  doi: 10.1016/S0065-3454(07)37003-4.  Google Scholar

[24]

A. Muthiah and R. Rajkumar, A novel algorithm for solving job-shop scheduling problem, Mechanika, 23 (2017), 610-617.  doi: 10.5755/j01.mech.23.4.14055.  Google Scholar

[25]

B. NaderiS. M. T. Fatemi GhomiM. Aminnayeri and M. Zandieh, Scheduling open shops with parallel machines to minimize total completion time, J. Comput. Appl. Math., 5 (2011), 1275-1287.  doi: 10.1016/j.cam.2010.08.013.  Google Scholar

[26]

Y. Nagata and I. Ono, A guided local search with iterative ejections of bottleneck operations for the job shop scheduling problem, Comput. Oper. Res., 90 (2018), 60-71.  doi: 10.1016/j.cor.2017.09.017.  Google Scholar

[27]

S. Ouadfel and A. Taleb-Ahmed, Social spiders optimization and flower pollination algorithm for multilevel image thresholding: A performance study, Expert Syst. with Applications, 55 (2016), 566-584.  doi: 10.1016/j.eswa.2016.02.024.  Google Scholar

[28]

B. PengZ. Lü and T. C. E. Cheng, A tabu search/path relinking algorithm to solve the job shop scheduling problem, Comput. Oper. Res., 53 (2015), 154-164.  doi: 10.1016/j.cor.2014.08.006.  Google Scholar

[29]

P. Pongchairerks, A Two-Level Metaheuristic Algorithm for the Job-Shop Scheduling Problem, Complexity, 1 (2019), 1-11.  doi: 10.1155/2019/8683472.  Google Scholar

[30]

A. Ponsich and C. A. Coello Coello, A hybrid Differential Evolution-Tabu Search algorithm for the solution of Job-Shop Scheduling Problems, Applied Soft Computing, 13 (2013), 462-474.  doi: 10.1016/j.asoc.2012.07.034.  Google Scholar

[31]

R. Qing-dao-er-ji and Y. Wang, A new hybrid genetic algorithm for job shop scheduling problem, Comput. Oper. Res., 39 (2012), 2291-2299.  doi: 10.1016/j.cor.2011.12.005.  Google Scholar

[32]

K. Rameshkumar and C. Rajendran, A novel discrete PSO algorithm for solving job shop scheduling problem to minimize makespan, IOP Conference Series: Materials Science and Engineering, 310 (2018), 21-43.  doi: 10.1088/1757-899X/310/1/012143.  Google Scholar

[33]

F. Ramezani and S. Lotfi, Social-Based Algorithm (SBA), Applied Soft Computing, 13 (2013), 2837-2856.  doi: 10.1016/j.asoc.2012.05.018.  Google Scholar

[34]

R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[35]

C. J. TanS. C. NeohC. P. LimS. HanounW. P. WongC. K. Loo and S. Nahavandi, Application of an evolutionary algorithm-based ensemble model to job-shop scheduling, J. of Intelligent Manufacturing, 30 (2019), 879-890.  doi: 10.1007/s10845-016-1291-1.  Google Scholar

[36]

R. F. Tavares Neto and M. Godinho Filho, Literature review regarding Ant Colony Optimization applied to scheduling problems: Guidelines for implementation and directions for future research, Engineering Applications of Artificial Intelligence, 26 (2013), 150-161.  doi: 10.1016/j.engappai.2012.03.011.  Google Scholar

[37]

W. Teekeng and A. Thammano, Modified genetic algorithm for flexible job-shop scheduling problems, Procedia Computer Science, 12 (2012), 122-128.  doi: 10.1016/j.procs.2012.09.041.  Google Scholar

[38]

C. M. Xiang, Observation on the flying habits of social spiders, Chinese J. of Zoology, 3 (1986), 11-11.   Google Scholar

[39]

L.-N. XingY.-W. ChenP. WangQ.-S. Zhao and J. Xiong, A knowledge-based ant colony optimization for flexible job shop scheduling problems, Applied Soft Computing, 10 (2010), 888-896.  doi: 10.1016/j.asoc.2009.10.006.  Google Scholar

[40]

R. YusofM. KhalidG. T. Hui and S. M. Yusof, Solving job shop scheduling problem using a hybrid parallel micro genetic algorithm, Applied Soft Computing, 11 (2011), 5782-5792.  doi: 10.1016/j.asoc.2011.01.046.  Google Scholar

[41]

R. Zhang and C. Wu, A simulated annealing algorithm based on block properties for the job shop scheduling problem with total weighted tardiness objective, Comput. Oper. Res., 38 (2011), 854-867.  doi: 10.1016/j.cor.2010.09.014.  Google Scholar

[42]

G. ZobolasC. D. Tarantilis and G. loannou, A hybrid evolutionary algorithm for the job shop scheduling problem, J. of the Oper. Res. Society, 60 (2009), 221-235.  doi: 10.1057/palgrave.jors.2602534.  Google Scholar

[43]

G. I. Zobolas, C. D. Tarantilis and G. Ioannou, Exact, heuristic and meta-heuristic algorithms for solving job shop scheduling problems, in Metaheuristics for Scheduling in Industrial and Manufacturing Applications, Studies in Computational Intelligence, 2, Springer, Berlin, 2008, 19–40. doi: 10.1007/978-3-540-78985-7_1.  Google Scholar

show all references

References:
[1]

R. F. Abdel-Kader, An improved PSO algorithm with genetic and neighborhood-based diversity operators for the job shop scheduling problem, Applied Artificial Intelligence, 32 (2018), 433-462.  doi: 10.1080/08839514.2018.1481903.  Google Scholar

[2]

M. Amirghasemi and R. Zamani, An effective asexual genetic algorithm for solving the job shop scheduling problem, Computers & Industrial Engineering, 83 (2015), 123-138.  doi: 10.1016/j.cie.2015.02.011.  Google Scholar

[3]

A. ElmiM. SolimanpurbS. Topaloglua and A. Elmic, A simulated annealing algorithm for the job shop cell scheduling problem with intercellular moves and reentrant parts, Computers & Industrial Engineering, 61 (2011), 171-178.  doi: 10.1016/j.cie.2011.03.007.  Google Scholar

[4]

J. E. Beasley, Or-library: Distributing test problems by electronic mail, J. of the Operational Research Society, 41 (1990), 1069-1072.  doi: 10.2307/2582903.  Google Scholar

[5]

E. Cuevas, M. A. Díaz Cortés and D. A. O. Navarro, Advances of Evolutionary Computation: Methods and Operators, Studies in Computational Intelligence, 629, Springer, 2016, 9–33. doi: 10.1007/978-3-319-28503-0.  Google Scholar

[6]

E. Cuevas, M. Cienfuegos, R. Rojas and A. Padilla, Computational Intelligence Applications in Modeling and Control, Studies in Computational Intelligence, 575, Springer, 2015, 123–146. doi: 10.1007/978-3-319-11017-2.  Google Scholar

[7]

E. CuevasM. CienfuegosD. Zaldivar and M. Perez-Cisneros, A swarm optimization algorithm inspired in the behavior of the social-spider, Expert Systems with Applications, 40 (2013), 6374-6384.  doi: 10.1016/j.eswa.2013.05.041.  Google Scholar

[8]

E. Cuevas, V. Osuna and D. Oliva, Evolutionary Computation Techniques: A Comparative Perspective, Studies in Computational Intelligence, 686 (2017), 65–93. doi: 10.1007/978-3-319-51109-2.  Google Scholar

[9]

T. K. DaoT. S. Pan and J. S. Pan, Parallel bat algorithm for optimizing makespan in job shop scheduling problems, J. of Intelligent Manufacturing, 29 (2018), 451-462.  doi: 10.1007/s10845-015-1121-x.  Google Scholar

[10]

N. FiǧlaliC. ÖzkaleO. Engin A. and Fi ǧlali, Investigation of Ant System parameter interactions by using design of experiments for job-shop scheduling problems, Computers & Industrial Engineering, 56 (2009), 538-559.  doi: 10.1016/j.cie.2007.06.001.  Google Scholar

[11]

H. Fisher and G. L. Thompson, Probabilistic learning combinations of local job-shop scheduling rules, in Industrial Scheduling, Prentice Hall, 1963, 225–251. Google Scholar

[12]

L. GaoX. LiX. WenC. Lu and F. Wen, A hybrid algorithm based on a new neighborhood structure evaluation method for job shop scheduling problem, Computers & Industrial Engineering, 88 (2015), 417-429.  doi: 10.1016/j.cie.2015.08.002.  Google Scholar

[13]

A. S. Jain and S. Meeran, Deterministic job-shop scheduling: Past, present and future, European J. of Operational Research, 113 (1999), 390-434.  doi: 10.1016/S0377-2217(98)00113-1.  Google Scholar

[14]

S. KavithaP. VenkumarN. Rajini and P. Pitchipoo, An efficient social spider optimization for flexible job shop scheduling problem, J. of Advanced Manufacturing Systems, 17 (2018), 181-196.  doi: 10.1142/S0219686718500117.  Google Scholar

[15]

M. Kurdi, A new hybrid island model genetic algorithm for job shop scheduling problem, Computers & Industrial Engineering, 88 (2015), 273-283.  doi: 10.1016/j.cie.2015.07.015.  Google Scholar

[16]

M. Kurdi, A Social Spider Optimization Algorithm for Hybrid Flow Shop Scheduling with Multiprocessor Task, 12th International NCM Conference: Challenges in Industrial Engineering & Operation Management, 2018. Google Scholar

[17]

M. Kurdi, An effective genetic algorithm with a critical-path-guided Giffler and Thompson crossover operator for job shop scheduling problem, International J. of Intelligent Systems and Applications in Engineering, 7 (2019), 13-18.  doi: 10.18201/ijisae.2019751247.  Google Scholar

[18]

M. Kurdi, An effective new island model genetic algorithm for job shop scheduling problem, Comput. Oper. Res., 67 (2016), 132-142.  doi: 10.1016/j.cor.2015.10.005.  Google Scholar

[19]

M. Kurdi, An improved island model memetic algorithm with a new cooperation phase for multi-objective job shop scheduling problem, Computers & Industrial Engineering, 111 (2017), 183-201.  doi: 10.1016/j.cie.2017.07.021.  Google Scholar

[20]

T.-L. LinS.-J. HorngT.-W. KaoY-.H. ChenR.-S. RunR.-J. ChenJ.-L. Lai and I.-H. Kuo, An efficient job-shop scheduling algorithm based on particle swarm optimization, Expert Systems with Applications, 37 (2010), 2629-2636.  doi: 10.1016/j.eswa.2009.08.015.  Google Scholar

[21]

M. LiuZ.-J. SunJ.-W. Yan and J.-S. Kang, An adaptive annealing genetic algorithm for the job-shop planning and scheduling problem, Expert Systems with Applications, 38 (2011), 9248-9255.  doi: 10.1016/j.eswa.2011.01.136.  Google Scholar

[22]

S. LuC. Sun and Z. Lu, An improved quantum-behaved particle swarm optimization method for short-term combined economic emission hydrothermal scheduling, Energy Conversion and Management, 51 (2010), 561-571.  doi: 10.1016/j.enconman.2009.10.024.  Google Scholar

[23]

T. B. Lubin, The Evolution of Sociality in Spiders, Advances in the Study of Behavior, 37 (2007), 83-145.  doi: 10.1016/S0065-3454(07)37003-4.  Google Scholar

[24]

A. Muthiah and R. Rajkumar, A novel algorithm for solving job-shop scheduling problem, Mechanika, 23 (2017), 610-617.  doi: 10.5755/j01.mech.23.4.14055.  Google Scholar

[25]

B. NaderiS. M. T. Fatemi GhomiM. Aminnayeri and M. Zandieh, Scheduling open shops with parallel machines to minimize total completion time, J. Comput. Appl. Math., 5 (2011), 1275-1287.  doi: 10.1016/j.cam.2010.08.013.  Google Scholar

[26]

Y. Nagata and I. Ono, A guided local search with iterative ejections of bottleneck operations for the job shop scheduling problem, Comput. Oper. Res., 90 (2018), 60-71.  doi: 10.1016/j.cor.2017.09.017.  Google Scholar

[27]

S. Ouadfel and A. Taleb-Ahmed, Social spiders optimization and flower pollination algorithm for multilevel image thresholding: A performance study, Expert Syst. with Applications, 55 (2016), 566-584.  doi: 10.1016/j.eswa.2016.02.024.  Google Scholar

[28]

B. PengZ. Lü and T. C. E. Cheng, A tabu search/path relinking algorithm to solve the job shop scheduling problem, Comput. Oper. Res., 53 (2015), 154-164.  doi: 10.1016/j.cor.2014.08.006.  Google Scholar

[29]

P. Pongchairerks, A Two-Level Metaheuristic Algorithm for the Job-Shop Scheduling Problem, Complexity, 1 (2019), 1-11.  doi: 10.1155/2019/8683472.  Google Scholar

[30]

A. Ponsich and C. A. Coello Coello, A hybrid Differential Evolution-Tabu Search algorithm for the solution of Job-Shop Scheduling Problems, Applied Soft Computing, 13 (2013), 462-474.  doi: 10.1016/j.asoc.2012.07.034.  Google Scholar

[31]

R. Qing-dao-er-ji and Y. Wang, A new hybrid genetic algorithm for job shop scheduling problem, Comput. Oper. Res., 39 (2012), 2291-2299.  doi: 10.1016/j.cor.2011.12.005.  Google Scholar

[32]

K. Rameshkumar and C. Rajendran, A novel discrete PSO algorithm for solving job shop scheduling problem to minimize makespan, IOP Conference Series: Materials Science and Engineering, 310 (2018), 21-43.  doi: 10.1088/1757-899X/310/1/012143.  Google Scholar

[33]

F. Ramezani and S. Lotfi, Social-Based Algorithm (SBA), Applied Soft Computing, 13 (2013), 2837-2856.  doi: 10.1016/j.asoc.2012.05.018.  Google Scholar

[34]

R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[35]

C. J. TanS. C. NeohC. P. LimS. HanounW. P. WongC. K. Loo and S. Nahavandi, Application of an evolutionary algorithm-based ensemble model to job-shop scheduling, J. of Intelligent Manufacturing, 30 (2019), 879-890.  doi: 10.1007/s10845-016-1291-1.  Google Scholar

[36]

R. F. Tavares Neto and M. Godinho Filho, Literature review regarding Ant Colony Optimization applied to scheduling problems: Guidelines for implementation and directions for future research, Engineering Applications of Artificial Intelligence, 26 (2013), 150-161.  doi: 10.1016/j.engappai.2012.03.011.  Google Scholar

[37]

W. Teekeng and A. Thammano, Modified genetic algorithm for flexible job-shop scheduling problems, Procedia Computer Science, 12 (2012), 122-128.  doi: 10.1016/j.procs.2012.09.041.  Google Scholar

[38]

C. M. Xiang, Observation on the flying habits of social spiders, Chinese J. of Zoology, 3 (1986), 11-11.   Google Scholar

[39]

L.-N. XingY.-W. ChenP. WangQ.-S. Zhao and J. Xiong, A knowledge-based ant colony optimization for flexible job shop scheduling problems, Applied Soft Computing, 10 (2010), 888-896.  doi: 10.1016/j.asoc.2009.10.006.  Google Scholar

[40]

R. YusofM. KhalidG. T. Hui and S. M. Yusof, Solving job shop scheduling problem using a hybrid parallel micro genetic algorithm, Applied Soft Computing, 11 (2011), 5782-5792.  doi: 10.1016/j.asoc.2011.01.046.  Google Scholar

[41]

R. Zhang and C. Wu, A simulated annealing algorithm based on block properties for the job shop scheduling problem with total weighted tardiness objective, Comput. Oper. Res., 38 (2011), 854-867.  doi: 10.1016/j.cor.2010.09.014.  Google Scholar

[42]

G. ZobolasC. D. Tarantilis and G. loannou, A hybrid evolutionary algorithm for the job shop scheduling problem, J. of the Oper. Res. Society, 60 (2009), 221-235.  doi: 10.1057/palgrave.jors.2602534.  Google Scholar

[43]

G. I. Zobolas, C. D. Tarantilis and G. Ioannou, Exact, heuristic and meta-heuristic algorithms for solving job shop scheduling problems, in Metaheuristics for Scheduling in Industrial and Manufacturing Applications, Studies in Computational Intelligence, 2, Springer, Berlin, 2008, 19–40. doi: 10.1007/978-3-540-78985-7_1.  Google Scholar

Figure 1.  FT20, Size = 100
Figure 2.  LA40, Size = 225
Figure 3.  ORB10, Size = 100
Figure 4.  YN04, Size = 400
Figure 5.  FT20, Size = 100
Figure 6.  LA40, Size = 225
Figure 7.  ORB10, Size = 100
Figure 8.  YN04, Size = 400
Table 1.  Notations for the JSSP
$ n $ the number of jobs
$ m $ the number of operations for one job
$ O_i $ the completion time of operation $ i $($ i=\{0, 1, 2, \dots, n*m+1\} $)
$ t_i $ the processing time of operation $ i $ on a given machine
$ \omega_{im} $ the flag of operation $ i $ initiated by machine $ m $
$ P_i $ all the predecessor operations of operation $ i $
$ A(t) $ the set of operations processed at time $ t $
$ o_{ji} $ the $ i $th operation of job $ j $
$ C_{\max} $ the makespan
$ n $ the number of jobs
$ m $ the number of operations for one job
$ O_i $ the completion time of operation $ i $($ i=\{0, 1, 2, \dots, n*m+1\} $)
$ t_i $ the processing time of operation $ i $ on a given machine
$ \omega_{im} $ the flag of operation $ i $ initiated by machine $ m $
$ P_i $ all the predecessor operations of operation $ i $
$ A(t) $ the set of operations processed at time $ t $
$ o_{ji} $ the $ i $th operation of job $ j $
$ C_{\max} $ the makespan
Table 2.  Simulation results for FT, LA, ORB and YN
Name Size($ n*m $) Algorithm Best Worst Mean Std.
FT20 20*5 PSO 1374.00 1521.00 1442.50 42.02
IGA 1744.00 2527.00 2025.50 198.95
DE 1456.00 1554.00 1506.00 27.64
SSO 1527.00 1527.00 1527.00 0
SSO-DM 1374.00 1374.00 1374.00 0
LA40 15*15 PSO 1498.00 1732.00 1576.05 59.79
IGA 2154.00 2803.00 2340.25 155.90
DE 1691.00 1824.00 1767.05 36.46
SSO 1834.00 1834.00 1834.00 0
SSO-DM 1528.00 1528.00 1528.00 0
ORB10 10*10 PSO 1039.00 1263.00 1150.05 48.84
IGA 1431.00 2121.00 1761.25 158.12
DE 1190.00 1293.00 1244.40 25.04
SSO 1345.00 1345.00 1345.00 0
SSO-DM 1114.00 1114.00 1114.00 0
YN4 20*20 PSO 1340.00 1607.00 1425.15 64.84
IGA 1826.00 2192.00 1997.90 116.48
DE 1486.00 1601.00 1570.75 26.15
SSO 1583.00 1583.00 1583.00 0
SSO-DM 1492.00 1492.00 1492.00 0
Name Size($ n*m $) Algorithm Best Worst Mean Std.
FT20 20*5 PSO 1374.00 1521.00 1442.50 42.02
IGA 1744.00 2527.00 2025.50 198.95
DE 1456.00 1554.00 1506.00 27.64
SSO 1527.00 1527.00 1527.00 0
SSO-DM 1374.00 1374.00 1374.00 0
LA40 15*15 PSO 1498.00 1732.00 1576.05 59.79
IGA 2154.00 2803.00 2340.25 155.90
DE 1691.00 1824.00 1767.05 36.46
SSO 1834.00 1834.00 1834.00 0
SSO-DM 1528.00 1528.00 1528.00 0
ORB10 10*10 PSO 1039.00 1263.00 1150.05 48.84
IGA 1431.00 2121.00 1761.25 158.12
DE 1190.00 1293.00 1244.40 25.04
SSO 1345.00 1345.00 1345.00 0
SSO-DM 1114.00 1114.00 1114.00 0
YN4 20*20 PSO 1340.00 1607.00 1425.15 64.84
IGA 1826.00 2192.00 1997.90 116.48
DE 1486.00 1601.00 1570.75 26.15
SSO 1583.00 1583.00 1583.00 0
SSO-DM 1492.00 1492.00 1492.00 0
Table 3.  Experimental results of running time of the algorithm
IGA PSO DE SSO SSO-DM
FT20 4.7350 1.3626 0.6521 0.7993 0.8725
LA40 4.2764 2.6175 1.0695 1.2537 1.2961
ORB10 2.7547 1.8100 1.1091 1.3127 1.0230
YN04 6.8851 3.0504 1.9461 2.3814 2.5842
IGA PSO DE SSO SSO-DM
FT20 4.7350 1.3626 0.6521 0.7993 0.8725
LA40 4.2764 2.6175 1.0695 1.2537 1.2961
ORB10 2.7547 1.8100 1.1091 1.3127 1.0230
YN04 6.8851 3.0504 1.9461 2.3814 2.5842
Table 4.  $ p $-values produced by Wilcoxon's test comparing SSO-DM vs. PSO, SSO-DM vs. IGA, SSO-DM vs. DE and SSO-DM vs. SSO, over the "average best-so-far" (Mean) values from Table 1 to Table 4
PSO IGA DE SSO
FT20 7.9772E-09 8.0065E-09 4.0136E-03 4.6826E-10
LA40 7.9918E-09 7.9918E-09 8.0065E-09 4.6826E-10
ORB10 7.9918E-09 8.0065E-09 7.9772E-09 4.6826E-10
YN04 2.0993E-07 8.0065E-09 4.0289E-02 4.6826E-10
PSO IGA DE SSO
FT20 7.9772E-09 8.0065E-09 4.0136E-03 4.6826E-10
LA40 7.9918E-09 7.9918E-09 8.0065E-09 4.6826E-10
ORB10 7.9918E-09 8.0065E-09 7.9772E-09 4.6826E-10
YN04 2.0993E-07 8.0065E-09 4.0289E-02 4.6826E-10
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