American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2021142
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Coordinated optimization of production scheduling and maintenance activities with machine reliability deterioration

Chaoming Hu 1, , Xiaofei Qian 1,, , Shaojun Lu 1,, , Xinbao Liu 1, and  Panos M Pardalos 2,

1. 

School of Management, Hefei University of Technology, 193 Tunxi Road, Hefei City, Anhui Province, China
2. 

Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, USA

* Corresponding author: Qian Xiaofei, Lu Shaojun

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: The first author is supported by the National Key Research and Development Program of China (2019YFB1705300)

In this paper, we investigate a coordinated optimization problem of production and maintenance where the machine reliability decreases with the use of the machine. Lower reliability means the machine is more likely to fail during the production stage. In the event of a machine failure, corrective maintenance (CM) of the machine is required, and the CM of the machine will cause a certain cost. Preventive maintenance (PM) can improve machine reliability and reduce machine failures during the production stage, but it will also cause a certain cost. To minimize the total maintenance cost, we must determine an appropriate PM plan to balance these two types of maintenance. In addition, the tardiness cost of jobs is also considered, which is affected not only by the processing sequence of jobs but also by the PM decision. The objective is to find the optimal job processing sequence and the optimal PM plan to minimize the total expected cost. To solve the proposed problem, an improved grey wolf optimizer (IGWO) algorithm is proposed. Experimental results show that the IGWO algorithm outperforms GA, VNS, TS, and standard GWO in optimization and computational stability.

Keywords: Production, maintenance, scheduling, reliability, IGWO algorithm.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Chaoming Hu, Xiaofei Qian, Shaojun Lu, Xinbao Liu, Panos M Pardalos. Coordinated optimization of production scheduling and maintenance activities with machine reliability deterioration. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021142
References:
[1]

J. Jaturonnatee, D. N. P. Murthy and R. Boondiskulchok, Optimal preventive maintenance of leased equipment with corrective minimal repairs, European Journal of Operational Research, 174 (2006), 201-215.  doi: 10.1016/j.ejor.2005.01.049.

[2]

K. Das, R. S. Lashkari and S. Sengupta, Machine reliability and preventive maintenance planning for cellular manufacturing systems, European Journal of Operational Research, 183 (2007), 162-180.  doi: 10.1016/j.ejor.2006.09.079.

[3]

H. Peng, Q. Feng and D. W. Coit, Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes, IIE transactions, 43 (2010), 12-22.  doi: 10.1080/0740817X.2010.491502.

[4]

C.-Y. Lee, Machine scheduling with an availability constraint, Optimization Applications in Scheduling Theory. J. Global Optim., 9 (1996), 395-416.  doi: 10.1007/BF00121681.

[5]

C. Low, M. Ji, C. -J.g Hsu and C.-T. Su, Minimizing the makespan in a single machine scheduling problems with flexible and periodic maintenance, Appl. Math. Model., 34 (2010), 334-342.  doi: 10.1016/j.apm.2009.04.014.

[6]

Y. Mati, Minimizing the makespan in the non-preemptive job-shop scheduling with limited machine availability, Computers & Industrial Engineering, 59 (2010), 537-543.  doi: 10.1016/j.cie.2010.06.010.

[7]

T. C. E. Cheng and Z. Liu, Approximability of two-machine no-wait flowshop scheduling with availability constraints, Oper. Res. Lett., 31 (2003), 319-322.  doi: 10.1016/S0167-6377(02)00230-4.

[8]

G. Li, M. Liu, S. P. Sethi and D. Xu, Parallel-machine scheduling with machine-dependent maintenance periodic recycles, International Journal of Production Economics, 186 (2017), 1-7.  doi: 10.1016/j.ijpe.2017.01.014.

[9]

M. Liu, S. Wang, C. 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.  doi: 10.1080/00207543.2015.1108535.

[10]

M. Kong, X. Liu, J. Pei, H. Cheng and P. M. Pardalos, A BRKGA-DE algorithm for parallel-batching scheduling with deterioration and learning effects on parallel machines under preventive maintenance consideration, Ann. Math. Artif. Intell., 88 (2020), 237-267.  doi: 10.1007/s10472-018-9602-1.

[11]

P. Perez-Gonzalez, V. Fernandez-Viagas and J. M. Framinan, Permutation flowshop scheduling with periodic maintenance and makespan objective, Computers & Industrial Engineering, 143 (2020), 106369. doi: 10.1016/j.cie.2020.106369.

[12]

A. Berrichi, F. Yalaoui, L. Amodeo and M. Mezghiche, Bi-objective ant colony optimization approach to optimize production and maintenance scheduling, Comput. Oper. Res., 37 (2010), 1584-1596.  doi: 10.1016/j.cor.2009.11.017.

[13]

H. Mokhtari, A. Mozdgir and I. N. Kamal Abadi, A reliability/availability approach to joint production and maintenance scheduling with multiple preventive maintenance services, International Journal of Production Research, 50 (2012), 5906-5925.  doi: 10.1080/00207543.2011.637092.

[14]

Z. Lu, W. Cui and X. Han, Integrated production and preventive maintenance scheduling for a single machine with failure uncertainty, Computers & Industrial Engineering, 80 (2015), 236-244.  doi: 10.1016/j.cie.2014.12.017.

[15]

R. Renmansour, H. Allaoui, A. Artiba, S. Iassinovski and R. Pellerin, Simulation-based approach to joint production and preventive maintenance scheduling on a failure-prone machine, Journal of Quality in Maintenance Engineering, 17 (2011). doi: 10.1108/13552511111157371.

[16]

Q. Liu, M. Dong and F. F. Chen, Single-machine-based joint optimization of predictive maintenance planning and production scheduling, Robotics and Computer-Integrated Manufacturing, 51 (2018), 238-247.  doi: 10.1016/j.rcim.2018.01.002.

[17]

Y. An, X. Chen, J. Zhang and Y. Li, A hybrid multi-objective evolutionary algorithm to integrate optimization of the production scheduling and imperfect cutting tool maintenance considering total energy consumption, Journal of Cleaner Production, 268 (2020), 121540. doi: 10.1016/j.jclepro.2020.121540.

[18]

H. Feng, L. Xi, L. Xiao, T. Xia and E. Pan, Imperfect preventive maintenance optimization for flexible flowshop manufacturing cells considering sequence-dependent group scheduling, Reliability Engineering & System Safety, 176 (2018), 218-229.  doi: 10.1016/j.ress.2018.04.004.

[19]

J. Hu and Z. Jiang, Job scheduling integrated with imperfect preventive maintenance considering time-varying operating condition, IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), (2017), 578–582. doi: 10.1109/IEEM.2017.8289957.

[20]

C. S. Wong, F. T. S. Chan and S. H. Chung, A joint production scheduling approach considering multiple resources and preventive maintenance tasks, International Journal of Production Research, 51 (2013), 883-896.  doi: 10.1080/00207543.2012.677070.

[21]

S. Kumar and B. K. Lad, Integrated production and maintenance planning for parallel machine system considering cost of rejection, Journal of the Operational Research Society, 68 (2017), 834-846.  doi: 10.1057/jors.2016.46.

[22]

S. Lu, X. Liu, J. Pei, M. T. Thai and P. M. Pardalos, A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity, Applied Soft Computing, 66 (2018), 168-182.  doi: 10.1016/j.asoc.2018.02.018.

[23]

D. Lin, M. J. Zuo and R. C. M. Yam, General sequential imperfect preventive maintenance models, International Journal of Reliability, Quality and Safety Engineering, 7 (2000), 253-266.  doi: 10.1142/S0218539300000213.

[24]

M.-C. Fitouhi and M. Nourelfath, Integrating noncyclical preventive maintenance scheduling and production planning for a single machine, International Journal of Production Economics, 136 (2012), 344-351.  doi: 10.1016/j.ijpe.2011.12.021.

[25]

S. Laohanan and D. Banjerdpongchai, Dynamic programming approach to optimal maintenance scheduling of substation equipment considering life cycle cost and reliability, International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON). IEEE, (2018), 388–391. doi: 10.1109/ECTICon.2018.8619884.

[26]

P. L. Ramos, D. C. Nascimento, C. Cocolo, et al., Reliability-centered maintenance: Analyzing failure in harvest sugarcane machine using some generalizations of the Weibull distribution, Modelling and Simulation in Engineering, (2018), 2018. doi: 10.1155/2018/1241856.

[27]

D. C. Idoniboyeobu, B. A. Wokoma and V. C. Ibanibo, Preventive maintenance for substation with aging equipment using weibull distribution, American Journal of Engineering Research, 7 (2018), 95-101. 

[28]

J. K. Lenstra, A. H. G. R. Kan and P. Brucker, Complexity of machine scheduling problems, Ann. of Discrete Math., North-Holland, Amsterdam, 1 (1977), 343-362. 

[29]

S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.  doi: 10.1016/j.advengsoft.2013.12.007.

[30]

G. M. Komaki and V. Kayvanfar, Grey Wolf Optimizer algorithm for the two-stage assembly flow shop scheduling problem with release time, Journal of Computational Science, 8 (2015), 109-120.  doi: 10.1016/j.jocs.2015.03.011.

[31]

C. Lu, L. Gao, X. Li and S. Xiao, A hybrid multi-objective grey wolf optimizer for dynamic scheduling in a real-world welding industry, Engineering Applications of Artificial Intelligence, 57 (2017), 61-79.  doi: 10.1016/j.engappai.2016.10.013.

[32]

M. K. Sattar, A. Ahmad, S. Fayyaz and et al., Ramp rate handling strategies in dynamic economic load dispatch (DELD) problem using grey wolf optimizer (GWO), Journal of the Chinese Institute of Engineers, 43 (2020), 200-213.  doi: 10.1080/02533839.2019.1694446.

[33]

A.-M. Golmohammadi, H. Bani-Asadi, H. J. Zanjani and H. Tikani, A genetic algorithm for preemptive scheduling of a single machine, Journal of the Chinese Institute of Engineers, 7 (2016), 607-614.  doi: 10.5267/j.ijiec.2016.3.004.

[34]

R. Logendran and A. Sonthinen, A Tabu search-based approach for scheduling job-shop type flexible manufacturing systems, Journal of the Operational Research Society, 48 (1997), 264-277.  doi: 10.1057/palgrave.jors.2600373.

[35]

D. Lei, Variable neighborhood search for two-agent flow shop scheduling problem, Computers & Industrial Engineering, 80 (2015), 125-131.  doi: 10.1016/j.cie.2014.11.024.

[36]

M. Zandieh, A. R. Khatami and S. H. A. Rahmati, Flexible job shop scheduling under condition-based maintenance: Improved version of imperialist competitive algorithm, Applied Soft Computing, 58 (2017), 449-464.  doi: 10.1016/j.asoc.2017.04.060.

show all references

References:
[1]

J. Jaturonnatee, D. N. P. Murthy and R. Boondiskulchok, Optimal preventive maintenance of leased equipment with corrective minimal repairs, European Journal of Operational Research, 174 (2006), 201-215.  doi: 10.1016/j.ejor.2005.01.049.

[2]

K. Das, R. S. Lashkari and S. Sengupta, Machine reliability and preventive maintenance planning for cellular manufacturing systems, European Journal of Operational Research, 183 (2007), 162-180.  doi: 10.1016/j.ejor.2006.09.079.

[3]

H. Peng, Q. Feng and D. W. Coit, Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes, IIE transactions, 43 (2010), 12-22.  doi: 10.1080/0740817X.2010.491502.

[4]

C.-Y. Lee, Machine scheduling with an availability constraint, Optimization Applications in Scheduling Theory. J. Global Optim., 9 (1996), 395-416.  doi: 10.1007/BF00121681.

[5]

C. Low, M. Ji, C. -J.g Hsu and C.-T. Su, Minimizing the makespan in a single machine scheduling problems with flexible and periodic maintenance, Appl. Math. Model., 34 (2010), 334-342.  doi: 10.1016/j.apm.2009.04.014.

[6]

Y. Mati, Minimizing the makespan in the non-preemptive job-shop scheduling with limited machine availability, Computers & Industrial Engineering, 59 (2010), 537-543.  doi: 10.1016/j.cie.2010.06.010.

[7]

T. C. E. Cheng and Z. Liu, Approximability of two-machine no-wait flowshop scheduling with availability constraints, Oper. Res. Lett., 31 (2003), 319-322.  doi: 10.1016/S0167-6377(02)00230-4.

[8]

G. Li, M. Liu, S. P. Sethi and D. Xu, Parallel-machine scheduling with machine-dependent maintenance periodic recycles, International Journal of Production Economics, 186 (2017), 1-7.  doi: 10.1016/j.ijpe.2017.01.014.

[9]

M. Liu, S. Wang, C. 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.  doi: 10.1080/00207543.2015.1108535.

[10]

M. Kong, X. Liu, J. Pei, H. Cheng and P. M. Pardalos, A BRKGA-DE algorithm for parallel-batching scheduling with deterioration and learning effects on parallel machines under preventive maintenance consideration, Ann. Math. Artif. Intell., 88 (2020), 237-267.  doi: 10.1007/s10472-018-9602-1.

[11]

P. Perez-Gonzalez, V. Fernandez-Viagas and J. M. Framinan, Permutation flowshop scheduling with periodic maintenance and makespan objective, Computers & Industrial Engineering, 143 (2020), 106369. doi: 10.1016/j.cie.2020.106369.

[12]

A. Berrichi, F. Yalaoui, L. Amodeo and M. Mezghiche, Bi-objective ant colony optimization approach to optimize production and maintenance scheduling, Comput. Oper. Res., 37 (2010), 1584-1596.  doi: 10.1016/j.cor.2009.11.017.

[13]

H. Mokhtari, A. Mozdgir and I. N. Kamal Abadi, A reliability/availability approach to joint production and maintenance scheduling with multiple preventive maintenance services, International Journal of Production Research, 50 (2012), 5906-5925.  doi: 10.1080/00207543.2011.637092.

[14]

Z. Lu, W. Cui and X. Han, Integrated production and preventive maintenance scheduling for a single machine with failure uncertainty, Computers & Industrial Engineering, 80 (2015), 236-244.  doi: 10.1016/j.cie.2014.12.017.

[15]

R. Renmansour, H. Allaoui, A. Artiba, S. Iassinovski and R. Pellerin, Simulation-based approach to joint production and preventive maintenance scheduling on a failure-prone machine, Journal of Quality in Maintenance Engineering, 17 (2011). doi: 10.1108/13552511111157371.

[16]

Q. Liu, M. Dong and F. F. Chen, Single-machine-based joint optimization of predictive maintenance planning and production scheduling, Robotics and Computer-Integrated Manufacturing, 51 (2018), 238-247.  doi: 10.1016/j.rcim.2018.01.002.

[17]

Y. An, X. Chen, J. Zhang and Y. Li, A hybrid multi-objective evolutionary algorithm to integrate optimization of the production scheduling and imperfect cutting tool maintenance considering total energy consumption, Journal of Cleaner Production, 268 (2020), 121540. doi: 10.1016/j.jclepro.2020.121540.

[18]

H. Feng, L. Xi, L. Xiao, T. Xia and E. Pan, Imperfect preventive maintenance optimization for flexible flowshop manufacturing cells considering sequence-dependent group scheduling, Reliability Engineering & System Safety, 176 (2018), 218-229.  doi: 10.1016/j.ress.2018.04.004.

[19]

J. Hu and Z. Jiang, Job scheduling integrated with imperfect preventive maintenance considering time-varying operating condition, IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), (2017), 578–582. doi: 10.1109/IEEM.2017.8289957.

[20]

C. S. Wong, F. T. S. Chan and S. H. Chung, A joint production scheduling approach considering multiple resources and preventive maintenance tasks, International Journal of Production Research, 51 (2013), 883-896.  doi: 10.1080/00207543.2012.677070.

[21]

S. Kumar and B. K. Lad, Integrated production and maintenance planning for parallel machine system considering cost of rejection, Journal of the Operational Research Society, 68 (2017), 834-846.  doi: 10.1057/jors.2016.46.

[22]

S. Lu, X. Liu, J. Pei, M. T. Thai and P. M. Pardalos, A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity, Applied Soft Computing, 66 (2018), 168-182.  doi: 10.1016/j.asoc.2018.02.018.

[23]

D. Lin, M. J. Zuo and R. C. M. Yam, General sequential imperfect preventive maintenance models, International Journal of Reliability, Quality and Safety Engineering, 7 (2000), 253-266.  doi: 10.1142/S0218539300000213.

[24]

M.-C. Fitouhi and M. Nourelfath, Integrating noncyclical preventive maintenance scheduling and production planning for a single machine, International Journal of Production Economics, 136 (2012), 344-351.  doi: 10.1016/j.ijpe.2011.12.021.

[25]

S. Laohanan and D. Banjerdpongchai, Dynamic programming approach to optimal maintenance scheduling of substation equipment considering life cycle cost and reliability, International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON). IEEE, (2018), 388–391. doi: 10.1109/ECTICon.2018.8619884.

[26]

P. L. Ramos, D. C. Nascimento, C. Cocolo, et al., Reliability-centered maintenance: Analyzing failure in harvest sugarcane machine using some generalizations of the Weibull distribution, Modelling and Simulation in Engineering, (2018), 2018. doi: 10.1155/2018/1241856.

[27]

D. C. Idoniboyeobu, B. A. Wokoma and V. C. Ibanibo, Preventive maintenance for substation with aging equipment using weibull distribution, American Journal of Engineering Research, 7 (2018), 95-101. 

[28]

J. K. Lenstra, A. H. G. R. Kan and P. Brucker, Complexity of machine scheduling problems, Ann. of Discrete Math., North-Holland, Amsterdam, 1 (1977), 343-362. 

[29]

S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.  doi: 10.1016/j.advengsoft.2013.12.007.

[30]

G. M. Komaki and V. Kayvanfar, Grey Wolf Optimizer algorithm for the two-stage assembly flow shop scheduling problem with release time, Journal of Computational Science, 8 (2015), 109-120.  doi: 10.1016/j.jocs.2015.03.011.

[31]

C. Lu, L. Gao, X. Li and S. Xiao, A hybrid multi-objective grey wolf optimizer for dynamic scheduling in a real-world welding industry, Engineering Applications of Artificial Intelligence, 57 (2017), 61-79.  doi: 10.1016/j.engappai.2016.10.013.

[32]

M. K. Sattar, A. Ahmad, S. Fayyaz and et al., Ramp rate handling strategies in dynamic economic load dispatch (DELD) problem using grey wolf optimizer (GWO), Journal of the Chinese Institute of Engineers, 43 (2020), 200-213.  doi: 10.1080/02533839.2019.1694446.

[33]

A.-M. Golmohammadi, H. Bani-Asadi, H. J. Zanjani and H. Tikani, A genetic algorithm for preemptive scheduling of a single machine, Journal of the Chinese Institute of Engineers, 7 (2016), 607-614.  doi: 10.5267/j.ijiec.2016.3.004.

[34]

R. Logendran and A. Sonthinen, A Tabu search-based approach for scheduling job-shop type flexible manufacturing systems, Journal of the Operational Research Society, 48 (1997), 264-277.  doi: 10.1057/palgrave.jors.2600373.

[35]

D. Lei, Variable neighborhood search for two-agent flow shop scheduling problem, Computers & Industrial Engineering, 80 (2015), 125-131.  doi: 10.1016/j.cie.2014.11.024.

[36]

M. Zandieh, A. R. Khatami and S. H. A. Rahmati, Flexible job shop scheduling under condition-based maintenance: Improved version of imperialist competitive algorithm, Applied Soft Computing, 58 (2017), 449-464.  doi: 10.1016/j.asoc.2017.04.060.

Figure 1.  Failure rate adjustment model
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Figure 2.  Schematic diagram of the machine age
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Figure 3.  An example to illustrate the encoding method
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Figure 4.  A comparison of two functions
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Figure 5.  Flow chart of IGWO algorithm
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Figure 6.  (a), (t) Convergence curvers for different instances.
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Figure 7(a).  Relative percentage deviations of AVE
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Figure 7(b) .  Relative percentage deviations of VAR
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Table 1.  Main features of previous studies and ours
Authors Production system Unexpected failures PM Objectives Solutions
Berrichi et al. (2010) Parallel machine - Perfect Minimize $ C_{max} $ as well as the system unavailability Multi-Objective Ant Colony Optimization approach
Mokhtari et al. (2012) Parallel machine - Multiple-level Minimize the total system unavailability Population-based variable neighborhood search
Liu et al.(2018) Single machine - Multiple-level Minimize total cost Genetic algorithm
Hu et al.(2017) Single machine - Imperfect Minimize the total cost Enumeration method
Lu et al. (2018) Parallel machine - Perfect Minimize makespan A hybrid ABC-TS algorithm
Wong et al.(2013) Parallel machine - Multiple-level Minimize the makespan Genetic algorithm
Kumar et al.(2017) Parallel machine - Perfect Minimize overall operations cost Simulation-based Genetic Algorithm
Feng et al.(2018) Flexible flowshop ✓ Imperfect Minimize the total cost Simulated annealing embedded genetic algorithm
Lu et al. (2015) Single machine ✓ Perfect System robustness and stability Genetic algorithm
Benmansour et al.(2011) Single machine ✓ Perfect Minimize the cost related to production and maintenance Heuristic algorithm
An et al.(2020) Flexible job-shop - Imperfect Minimize the makespan and total cost Evolutionary algorithm
This paper Single machine ✓ Imperfect Minimize the total expected cost Improved grey wolf optimizer algorithm
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Download as excel

Authors Production system Unexpected failures PM Objectives Solutions
Berrichi et al. (2010) Parallel machine - Perfect Minimize $ C_{max} $ as well as the system unavailability Multi-Objective Ant Colony Optimization approach
Mokhtari et al. (2012) Parallel machine - Multiple-level Minimize the total system unavailability Population-based variable neighborhood search
Liu et al.(2018) Single machine - Multiple-level Minimize total cost Genetic algorithm
Hu et al.(2017) Single machine - Imperfect Minimize the total cost Enumeration method
Lu et al. (2018) Parallel machine - Perfect Minimize makespan A hybrid ABC-TS algorithm
Wong et al.(2013) Parallel machine - Multiple-level Minimize the makespan Genetic algorithm
Kumar et al.(2017) Parallel machine - Perfect Minimize overall operations cost Simulation-based Genetic Algorithm
Feng et al.(2018) Flexible flowshop ✓ Imperfect Minimize the total cost Simulated annealing embedded genetic algorithm
Lu et al. (2015) Single machine ✓ Perfect System robustness and stability Genetic algorithm
Benmansour et al.(2011) Single machine ✓ Perfect Minimize the cost related to production and maintenance Heuristic algorithm
An et al.(2020) Flexible job-shop - Imperfect Minimize the makespan and total cost Evolutionary algorithm
This paper Single machine ✓ Imperfect Minimize the total expected cost Improved grey wolf optimizer algorithm
Table 2.  Notations and their explanations
$ N $ Number of jobs
$ p_j $ The processing time of job $ j $, $ j = 1,…,N $
$ d_j $ The due date of job $ j $, $ j = 1,…,N $
$ w_j $ Tardiness cost per unit of time of job $ j $, $ j = 1,…,N $
$ c_l $ The unit basic cost of the factory.
$ t_r $ The time required for performing corrective maintenance (CM).
$ c_r $ The cost of a CM activity.
$ t_p $ The time required for performing preventive maintenance (PM).
$ c_p $ The cost of a PM activity.
$ x_{[i]j} $ Job sequencing decision variable, and it is a zero and one variable
$ y_{[i]} $ PM decision variable, and it is a zero and one variable
$ p_{[i]} $ The processing time of the $ i $th job in the sequence
$ f_{[i]}(t) $ The failure rate function during the processing of the $ i $th job and $ t $ is the machine age (i.e. the time since the last PM)
$ m $ The number of PMs performed on the machine before time 0.
$ Z_{[0]}^{f} $ The machine age at time 0.
$ Z_{[i]}^{s} $ The machine age when starting the processing of the $ i $th job
$ Z_{[i]}^{f} $ The machine age when finishing the processing of the $ i $th job
$ E(N_{[i]}) $ The expected number of failures during the processing of the $ i $th job
$ E(C_{[i]}) $ The expected completion time of the $ i $th job
$ E(T_{[i]}) $ The expected tardiness time of the $ i $th job
$ E(TC) $ The expected total cost.
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$ N $ Number of jobs
$ p_j $ The processing time of job $ j $, $ j = 1,…,N $
$ d_j $ The due date of job $ j $, $ j = 1,…,N $
$ w_j $ Tardiness cost per unit of time of job $ j $, $ j = 1,…,N $
$ c_l $ The unit basic cost of the factory.
$ t_r $ The time required for performing corrective maintenance (CM).
$ c_r $ The cost of a CM activity.
$ t_p $ The time required for performing preventive maintenance (PM).
$ c_p $ The cost of a PM activity.
$ x_{[i]j} $ Job sequencing decision variable, and it is a zero and one variable
$ y_{[i]} $ PM decision variable, and it is a zero and one variable
$ p_{[i]} $ The processing time of the $ i $th job in the sequence
$ f_{[i]}(t) $ The failure rate function during the processing of the $ i $th job and $ t $ is the machine age (i.e. the time since the last PM)
$ m $ The number of PMs performed on the machine before time 0.
$ Z_{[0]}^{f} $ The machine age at time 0.
$ Z_{[i]}^{s} $ The machine age when starting the processing of the $ i $th job
$ Z_{[i]}^{f} $ The machine age when finishing the processing of the $ i $th job
$ E(N_{[i]}) $ The expected number of failures during the processing of the $ i $th job
$ E(C_{[i]}) $ The expected completion time of the $ i $th job
$ E(T_{[i]}) $ The expected tardiness time of the $ i $th job
$ E(TC) $ The expected total cost.
Table 3.  Local search strategy
Step 1 Set the maximum number of searches $ s_{max} $ and let $ s = 0 $
Step 2 Randomly generate three unequal integer $ a,b,c $ in $ [1,N] $
Step 3 Let $ t = x_a, x_a = x_b, x_b = t $, and $ x_c = - x_c $ and then obtain a new solution $ X^{'} $
Step 4 Calculate the fitness of $ X^{'} $, and let $ s = s+1 $
Step 5 Judge whether $ f(X^)<f(X) $ is true, if so, let $ X \leftarrow X^{'} $ and stop the search, otherwise, go to step 6
Step 6 Judge whether $ s\leq or \le s_{max} $, if so, turn to Step 2, otherwise keep $ X $ and stop search
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Step 1 Set the maximum number of searches $ s_{max} $ and let $ s = 0 $
Step 2 Randomly generate three unequal integer $ a,b,c $ in $ [1,N] $
Step 3 Let $ t = x_a, x_a = x_b, x_b = t $, and $ x_c = - x_c $ and then obtain a new solution $ X^{'} $
Step 4 Calculate the fitness of $ X^{'} $, and let $ s = s+1 $
Step 5 Judge whether $ f(X^)<f(X) $ is true, if so, let $ X \leftarrow X^{'} $ and stop the search, otherwise, go to step 6
Step 6 Judge whether $ s\leq or \le s_{max} $, if so, turn to Step 2, otherwise keep $ X $ and stop search
Table 4.  Pseudocode of IGWO algorithm
1 Initializes the grey wolf population
2 Initializes the max number of iterations $ t_{max} $ and the local search probability $ \varepsilon $, and set $ a = 2 $, $ k = 1 $
3 Calculate the fitness of all individuals
4 Set the three individuals with the best fitness as $ X_\alpha $, $ X_\beta $, and $ X_\gamma $
5 while($ k<t_{max} $)
6     for each search agent
7       generate a random probability $ p $
8       if $ p\geq \varepsilon $
9         generate two random numbers $ r_1 $, $ r_2 $ in $ (0,1) $
10         calculate the coefficients $ A $, $ C $ by formula (4.3)-(4.4)
11         update the position of the current search agent by formula (4.5)-(4.7)
12       else if
13        update the position of the current search agent by Local search strategy
14       end if
15     update the population
16     calculate the fitness of all individuals
17     update $ X_\alpha $, $ X_\beta $, and $ X_\gamma $
18     update $ a $ by formula(4.8)
19     $ k = k+1 $
20 end while
21 rerurn $ X_\alpha $
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1 Initializes the grey wolf population
2 Initializes the max number of iterations $ t_{max} $ and the local search probability $ \varepsilon $, and set $ a = 2 $, $ k = 1 $
3 Calculate the fitness of all individuals
4 Set the three individuals with the best fitness as $ X_\alpha $, $ X_\beta $, and $ X_\gamma $
5 while($ k<t_{max} $)
6     for each search agent
7       generate a random probability $ p $
8       if $ p\geq \varepsilon $
9         generate two random numbers $ r_1 $, $ r_2 $ in $ (0,1) $
10         calculate the coefficients $ A $, $ C $ by formula (4.3)-(4.4)
11         update the position of the current search agent by formula (4.5)-(4.7)
12       else if
13        update the position of the current search agent by Local search strategy
14       end if
15     update the population
16     calculate the fitness of all individuals
17     update $ X_\alpha $, $ X_\beta $, and $ X_\gamma $
18     update $ a $ by formula(4.8)
19     $ k = k+1 $
20 end while
21 rerurn $ X_\alpha $
Table 5.  Experimental factors and their levels
The total number of jobs, $ N $ 10, 20, 30, …, 180,190,200
The failure rate adjustment parameter, $ \theta $ 1.05
The failure rate parameter, $ \beta $ 2
The failure rate parameter, $ \eta $ 50
The processing time of job $ j $, $ p_j $ Uniform$ [1,10] $
The due date of job $ j $, $ d_j $ Uniform $ [1,100] $
The tardiness cost per unit of time cost of job $ j $, $ w_j $ Uniform $ [0, 1] $
The unit production cost, $ c_l $ 1
The time required for performing corrective maintenance, $ t_r $ 1
The cost of a corrective maintenance activity, $ c_r $ 1
The time required for performing preventive maintenance, $ t_p $ 2
The cost of a preventive maintenance activity, $ c_p $ 2
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The total number of jobs, $ N $ 10, 20, 30, …, 180,190,200
The failure rate adjustment parameter, $ \theta $ 1.05
The failure rate parameter, $ \beta $ 2
The failure rate parameter, $ \eta $ 50
The processing time of job $ j $, $ p_j $ Uniform$ [1,10] $
The due date of job $ j $, $ d_j $ Uniform $ [1,100] $
The tardiness cost per unit of time cost of job $ j $, $ w_j $ Uniform $ [0, 1] $
The unit production cost, $ c_l $ 1
The time required for performing corrective maintenance, $ t_r $ 1
The cost of a corrective maintenance activity, $ c_r $ 1
The time required for performing preventive maintenance, $ t_p $ 2
The cost of a preventive maintenance activity, $ c_p $ 2
Table 6.  Comparison of GWO and IGWO
GWO IGWO
Instance MIN MAX AVE VAR MIN MAX AVE VAR
N = 10 72 78 76 0.87 65 67 66 0.28
N = 20 315 355 338 124 244 262 253 28
N = 30 595 641 615 263 420 479 442 198
N = 40 1595 1680 1649 3668 1330 1489 1424 11075
N = 50 2462 2683 2610 2375 2311 2518 2405 2231
N = 60 3608 4046 3849 14612 3171 3336 3266 2206
N = 70 4888 5268 5057 17758 4481 4722 4583 5833
N = 80 6760 7107 6929 20810 5909 6339 6089 9859
N = 90 7037 7328 7174 13697 6179 6544 6357 9622
N = 100 10781 11429 11147 35652 9768 10393 10101 33610
N = 110 13256 14317 13811 103514 12511 13184 12788 53694
N = 120 17281 18388 17783 106493 15711 16953 16331 104363
N = 130 20102 21580 20758 214099 18108 19249 18573 63512
N = 140 20919 22114 21594 112116 18600 19460 19020 20809
N = 150 23002 24554 23793 246926 19369 20752 19891 56352
N = 160 29341 30523 29801 157775 25098 25768 25436 61450
N = 170 34321 36133 35142 420426 28998 30418 29632 134003
N = 180 38393 40478 39255 390366 32914 34377 33557 177787
N = 190 49190 51420 50013 395090 42772 44139 43557 177696
N = 200 50416 52568 51498 575500 43267 44589 44003 102377
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GWO IGWO
Instance MIN MAX AVE VAR MIN MAX AVE VAR
N = 10 72 78 76 0.87 65 67 66 0.28
N = 20 315 355 338 124 244 262 253 28
N = 30 595 641 615 263 420 479 442 198
N = 40 1595 1680 1649 3668 1330 1489 1424 11075
N = 50 2462 2683 2610 2375 2311 2518 2405 2231
N = 60 3608 4046 3849 14612 3171 3336 3266 2206
N = 70 4888 5268 5057 17758 4481 4722 4583 5833
N = 80 6760 7107 6929 20810 5909 6339 6089 9859
N = 90 7037 7328 7174 13697 6179 6544 6357 9622
N = 100 10781 11429 11147 35652 9768 10393 10101 33610
N = 110 13256 14317 13811 103514 12511 13184 12788 53694
N = 120 17281 18388 17783 106493 15711 16953 16331 104363
N = 130 20102 21580 20758 214099 18108 19249 18573 63512
N = 140 20919 22114 21594 112116 18600 19460 19020 20809
N = 150 23002 24554 23793 246926 19369 20752 19891 56352
N = 160 29341 30523 29801 157775 25098 25768 25436 61450
N = 170 34321 36133 35142 420426 28998 30418 29632 134003
N = 180 38393 40478 39255 390366 32914 34377 33557 177787
N = 190 49190 51420 50013 395090 42772 44139 43557 177696
N = 200 50416 52568 51498 575500 43267 44589 44003 102377
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