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Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints
Department of Industrial Engineering and Management, Mazandaran University of Science and Technology, Babol, Iran |
In today's competitive world, scheduling problems are one of the most important and vital issues. In this study, a bi-objective unrelated parallel machine scheduling problem with worker allocation, sequence dependent setup times, precedence constraints, and machine eligibility is presented. The objective functions are to minimize the costs of tardiness and hiring workers. In order to formulate the proposed problem, a mixed-integer quadratic programming model is presented. A strategy called repair is also proposed to implement the precedence constraints. Because the problem is NP-hard, two metaheuristic algorithms, a multi-objective tabu search (MOTS) and a multi-objective simulated annealing (MOSA), are presented to tackle the problem. Furthermore, a hybrid metaheuristic algorithm is also developed. Finally, computational experiments are carried out to evaluate different test problems, and analysis of variance is done to compare the performance of the proposed algorithms. The results show that MOTS is doing better in terms of objective values and mean ideal distance (MID) metric, while the proposed hybrid algorithm outperforms in most cases, considering other employed comparison metrics.
References:
[1] |
M. Afzalirad and J. Rezaeian,
Resource-constrained unrelated parallel machine scheduling problem with sequence dependent setup times, precedence constraints and machine eligibility restrictions, Computers & Industrial Engineering, 98 (2016), 40-52.
doi: 10.1016/j.cie.2016.05.020. |
[2] |
M. Afzalirad and M. Shafipour,
Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions, Journal of Intelligent Manufacturing, 29 (2018), 423-437.
doi: 10.1007/s10845-015-1117-6. |
[3] |
O. A. Arik and M. D. Toksari,
Multi-objective fuzzy parallel machine scheduling problems under fuzzy job deterioration and learning effects, International Journal of Production Research, 56 (2018), 2488-2505.
doi: 10.1080/00207543.2017.1388932. |
[4] |
A. Baykasoglu,
Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy, International Journal for Numerical Methods in Engineering, 65 (2006), 406-424.
doi: 10.1002/nme.1455. |
[5] |
T. Çakar, R. Köker and Y. Sari,
Parallel robot scheduling to minimize mean tardiness with unequal release date and precedence constraints using a hybrid intelligent system, International Journal of Advanced Robotic Systems, 9 (2012), 252.
doi: 10.5772/54381. |
[6] |
C. L. Chen,
Iterated hybrid metaheuristic algorithms for unrelated parallel machines problem with unequal ready times and sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 60 (2012), 693-705.
doi: 10.1007/s00170-011-3623-9. |
[7] |
L. P. Cota, F. G. Guimarães, R. G. Ribeiro, I. R. Meneghini, F. B. de Oliveira, M. J. Souza and P. Siarry,
An adaptive multi-objective algorithm based on decomposition and large neighborhood search for a green machine scheduling problem, Swarm and Evolutionary Computation, 51 (2019), 100601.
doi: 10.1016/j.swevo.2019.100601. |
[8] |
M. Dhiflaoui, H. E. Nouri and O. B. Driss,
Dual-resource constraints in classical and flexible job shop problems: A state-of-the-art review, Procedia Computer Science, 126 (2018), 1507-1515.
doi: 10.1016/j.procs.2018.08.123. |
[9] |
E. C. H. Dorion, J. C. F. Guimarães, E. A. Severo, Z. C. Reis and P. M. Olea, Innovation and production management through a just in sequence strategy in a multinational brazilian metal-mechanic industry, 2014 IEEE International Conference on Management of Innovation and Technology, (2014), 54–60.
doi: 10.1109/ICMIT.2014.6942400. |
[10] |
P. Engrand, A multi-objective optimization approach based on simulated annealing and its application to nuclear fuel management, 1998. |
[11] |
A. E. Ezugwu, O. J. Adeleke and S. Viriri, Symbiotic organisms search algorithm for the unrelated parallel machines scheduling with sequence-dependent setup times, PloS One, 13 (2018), e0200030.
doi: 10.1371/journal.pone.0200030. |
[12] |
A. E. Ezugwu and F. Akutsah,
An improved firefly algorithm for the unrelated parallel machines scheduling problem with sequence-dependent setup times, IEEE Access, 6 (2018), 54459-54478.
doi: 10.1109/ACCESS.2018.2872110. |
[13] |
C. M. Fonseca, L. Paquete and M. López-Ibánez, An improved dimension-sweep algorithm for the hypervolume indicator, 2006 IEEE International Conference on Evolutionary Computation, (2006), 1157–1163.
doi: 10.1109/CEC.2006.1688440. |
[14] |
F. Glover,
Future paths for integer programming and links to artificial intelligence, Comput. Oper. Res., 13 (1986), 533-549.
doi: 10.1016/0305-0548(86)90048-1. |
[15] |
F. W. Glover and G. A. Kochenberger. (Eds.), Handbook of Metaheuristics, International Series in Operations Research & Management Science, 57. Kluwer Academic Publishers, Boston, MA, 2003. |
[16] |
G. Gong, R. Chiong, Q. Deng and X. Gong,
A hybrid artificial bee colony algorithm for flexible job shop scheduling with worker flexibility, International Journal of Production Research, 58 (2020), 4406-4420.
doi: 10.1080/00207543.2019.1653504. |
[17] |
G. Gong, R. Chiong, Q. Deng, W. Han, L. Zhang, W. Lin and K. Li,
Energy-efficient flexible flow shop scheduling with worker flexibility, Expert Systems with Applications, 141 (2020), 112902.
doi: 10.1016/j.eswa.2019.112902. |
[18] |
A. Hamzadayi and G. Yildiz,
Modeling and solving static m identical parallel machines scheduling problem with a common server and sequence dependent setup times, Computers & Industrial Engineering, 106 (2017), 287-298.
doi: 10.1016/j.cie.2017.02.013. |
[19] |
M. P. Hansen, Tabu search for multiobjective optimization: MOTS, Proceedings of the 13th International Conference on Multiple Criteria Decision Making, (1997), 574–586. |
[20] |
O. Hurdies,
Just in time (JIT) production: An effective approach to efficiency, Logistics & Supply Chain Review, 1 (2020), 32-39.
|
[21] |
B. Van Khanh and N. Van Hop,
Genetic algorithm with initial sequence for parallel machines scheduling with sequence dependent setup times based on earliness-tardiness, Journal of Industrial and Production Engineering, 38 (2021), 18-28.
doi: 10.1080/21681015.2020.1829111. |
[22] |
J. G. Kim, S. Song and B. Jeong,
Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times, International Journal of Production Research, 58 (2020), 1628-1643.
doi: 10.1080/00207543.2019.1672900. |
[23] |
S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi,
Optimization by simulated annealing, Science, 220 (1983), 671-680.
doi: 10.1126/science.220.4598.671. |
[24] |
D. Lei, Y. Yuan, J. Cai and D. Bai,
An imperialist competitive algorithm with memory for distributed unrelated parallel machines scheduling, International Journal of Production Research, 58 (2020), 597-614.
doi: 10.1080/00207543.2019.1598596. |
[25] |
H. M. Md, Lead time reduction and process cycle improvement of an ice-cream manufacturing factory in bangladesh by using value stream map and kanban board, Australian Journal Of Basic and Applied Sciences, (2016). |
[26] |
A. Munoz-Villamizar, J. Santos, J. Montoya-Torres and M. Alvaréz,
Improving effectiveness of parallel machine scheduling with earliness and tardiness costs: A case study, International Journal of Industrial Engineering Computations, 10 (2019), 375-392.
doi: 10.5267/j.ijiec.2019.2.001. |
[27] |
J. Rezaeian, N. Derakhshan, I. Mahdavi and R. A. Foroutan,
Due date assignment and JIT scheduling problem in blocking hybrid flow shop robotic cells with multiple robots and batch delivery cost, International Journal of Industrial Mathematics, 13 (2021), 145-162.
|
[28] |
M. Savsar,
Simulation analysis of a pull-push system for an electronic assembly line, International Journal of Production Economics, 51 (1997), 205-214.
doi: 10.1016/S0925-5273(97)00055-8. |
[29] |
G. Taguchi, Introduction to quality engineering: Designing quality into products and processes, No. 658.562 T3, (1986). |
[30] |
E. Vallada and R. Ruiz,
A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times, European J. Oper. Res., 211 (2011), 612-622.
doi: 10.1016/j.ejor.2011.01.011. |
[31] |
B. Wang and H. Wang, Multiobjective order acceptance and scheduling on unrelated parallel machines with machine eligibility constraints, Math. Probl. Eng., 2018 (2018), 12pp.
doi: 10.1155/2018/6024631. |
[32] |
I. L. Wang, Y. C. Wang and C. W. Chen,
Scheduling unrelated parallel machines in semiconductor manufacturing by problem reduction and local search heuristics, Flexible Services and Manufacturing Journal, 25 (2013), 343-366.
doi: 10.1007/s10696-012-9150-7. |
[33] |
S. Wang, X. Wang, J. Yu, S. Ma and M. Liu,
Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan, Journal of Cleaner Production, 193 (2018), 424-440.
doi: 10.1016/j.jclepro.2018.05.056. |
[34] |
J. Xu, X. Xu and S. Q. Xie,
Recent developments in Dual Resource Constrained (DRC) system research, European Journal of Operational Research, 215 (2011), 309-318.
doi: 10.1016/j.ejor.2011.03.004. |
[35] |
K. C. Ying and S. W. Lin,
Unrelated parallel machine scheduling with sequence-and machine-dependent setup times and due date constraints, International Journal of Innovative Computing, 8 (2012), 3279-3297.
|
[36] |
A. Zhang, X. Qi and G. Li,
Machine scheduling with soft precedence constraints, European J. Oper. Res., 282 (2020), 491-505.
doi: 10.1016/j.ejor.2019.09.041. |
[37] |
J. R. Zeidi and S. MohammadHosseini,
Scheduling unrelated parallel machines with sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 81 (2015), 1487-1496.
doi: 10.1007/s00170-015-7215-y. |
[38] |
L. Zhang, Q. Deng, G. Gong and W. Han,
A new unrelated parallel machine scheduling problem with tool changes to minimise the total energy consumption, International Journal of Production Research, 58 (2020), 6826-6845.
doi: 10.1080/00207543.2019.1685708. |
[39] |
Z. Zhu and X. Zhou,
An efficient evolutionary grey wolf optimizer for multi-objective flexible job shop scheduling problem with hierarchical job precedence constraints, Computers & Industrial Engineering, 140 (2020), 106280.
doi: 10.1016/j.cie.2020.106280. |
show all references
References:
[1] |
M. Afzalirad and J. Rezaeian,
Resource-constrained unrelated parallel machine scheduling problem with sequence dependent setup times, precedence constraints and machine eligibility restrictions, Computers & Industrial Engineering, 98 (2016), 40-52.
doi: 10.1016/j.cie.2016.05.020. |
[2] |
M. Afzalirad and M. Shafipour,
Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions, Journal of Intelligent Manufacturing, 29 (2018), 423-437.
doi: 10.1007/s10845-015-1117-6. |
[3] |
O. A. Arik and M. D. Toksari,
Multi-objective fuzzy parallel machine scheduling problems under fuzzy job deterioration and learning effects, International Journal of Production Research, 56 (2018), 2488-2505.
doi: 10.1080/00207543.2017.1388932. |
[4] |
A. Baykasoglu,
Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy, International Journal for Numerical Methods in Engineering, 65 (2006), 406-424.
doi: 10.1002/nme.1455. |
[5] |
T. Çakar, R. Köker and Y. Sari,
Parallel robot scheduling to minimize mean tardiness with unequal release date and precedence constraints using a hybrid intelligent system, International Journal of Advanced Robotic Systems, 9 (2012), 252.
doi: 10.5772/54381. |
[6] |
C. L. Chen,
Iterated hybrid metaheuristic algorithms for unrelated parallel machines problem with unequal ready times and sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 60 (2012), 693-705.
doi: 10.1007/s00170-011-3623-9. |
[7] |
L. P. Cota, F. G. Guimarães, R. G. Ribeiro, I. R. Meneghini, F. B. de Oliveira, M. J. Souza and P. Siarry,
An adaptive multi-objective algorithm based on decomposition and large neighborhood search for a green machine scheduling problem, Swarm and Evolutionary Computation, 51 (2019), 100601.
doi: 10.1016/j.swevo.2019.100601. |
[8] |
M. Dhiflaoui, H. E. Nouri and O. B. Driss,
Dual-resource constraints in classical and flexible job shop problems: A state-of-the-art review, Procedia Computer Science, 126 (2018), 1507-1515.
doi: 10.1016/j.procs.2018.08.123. |
[9] |
E. C. H. Dorion, J. C. F. Guimarães, E. A. Severo, Z. C. Reis and P. M. Olea, Innovation and production management through a just in sequence strategy in a multinational brazilian metal-mechanic industry, 2014 IEEE International Conference on Management of Innovation and Technology, (2014), 54–60.
doi: 10.1109/ICMIT.2014.6942400. |
[10] |
P. Engrand, A multi-objective optimization approach based on simulated annealing and its application to nuclear fuel management, 1998. |
[11] |
A. E. Ezugwu, O. J. Adeleke and S. Viriri, Symbiotic organisms search algorithm for the unrelated parallel machines scheduling with sequence-dependent setup times, PloS One, 13 (2018), e0200030.
doi: 10.1371/journal.pone.0200030. |
[12] |
A. E. Ezugwu and F. Akutsah,
An improved firefly algorithm for the unrelated parallel machines scheduling problem with sequence-dependent setup times, IEEE Access, 6 (2018), 54459-54478.
doi: 10.1109/ACCESS.2018.2872110. |
[13] |
C. M. Fonseca, L. Paquete and M. López-Ibánez, An improved dimension-sweep algorithm for the hypervolume indicator, 2006 IEEE International Conference on Evolutionary Computation, (2006), 1157–1163.
doi: 10.1109/CEC.2006.1688440. |
[14] |
F. Glover,
Future paths for integer programming and links to artificial intelligence, Comput. Oper. Res., 13 (1986), 533-549.
doi: 10.1016/0305-0548(86)90048-1. |
[15] |
F. W. Glover and G. A. Kochenberger. (Eds.), Handbook of Metaheuristics, International Series in Operations Research & Management Science, 57. Kluwer Academic Publishers, Boston, MA, 2003. |
[16] |
G. Gong, R. Chiong, Q. Deng and X. Gong,
A hybrid artificial bee colony algorithm for flexible job shop scheduling with worker flexibility, International Journal of Production Research, 58 (2020), 4406-4420.
doi: 10.1080/00207543.2019.1653504. |
[17] |
G. Gong, R. Chiong, Q. Deng, W. Han, L. Zhang, W. Lin and K. Li,
Energy-efficient flexible flow shop scheduling with worker flexibility, Expert Systems with Applications, 141 (2020), 112902.
doi: 10.1016/j.eswa.2019.112902. |
[18] |
A. Hamzadayi and G. Yildiz,
Modeling and solving static m identical parallel machines scheduling problem with a common server and sequence dependent setup times, Computers & Industrial Engineering, 106 (2017), 287-298.
doi: 10.1016/j.cie.2017.02.013. |
[19] |
M. P. Hansen, Tabu search for multiobjective optimization: MOTS, Proceedings of the 13th International Conference on Multiple Criteria Decision Making, (1997), 574–586. |
[20] |
O. Hurdies,
Just in time (JIT) production: An effective approach to efficiency, Logistics & Supply Chain Review, 1 (2020), 32-39.
|
[21] |
B. Van Khanh and N. Van Hop,
Genetic algorithm with initial sequence for parallel machines scheduling with sequence dependent setup times based on earliness-tardiness, Journal of Industrial and Production Engineering, 38 (2021), 18-28.
doi: 10.1080/21681015.2020.1829111. |
[22] |
J. G. Kim, S. Song and B. Jeong,
Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times, International Journal of Production Research, 58 (2020), 1628-1643.
doi: 10.1080/00207543.2019.1672900. |
[23] |
S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi,
Optimization by simulated annealing, Science, 220 (1983), 671-680.
doi: 10.1126/science.220.4598.671. |
[24] |
D. Lei, Y. Yuan, J. Cai and D. Bai,
An imperialist competitive algorithm with memory for distributed unrelated parallel machines scheduling, International Journal of Production Research, 58 (2020), 597-614.
doi: 10.1080/00207543.2019.1598596. |
[25] |
H. M. Md, Lead time reduction and process cycle improvement of an ice-cream manufacturing factory in bangladesh by using value stream map and kanban board, Australian Journal Of Basic and Applied Sciences, (2016). |
[26] |
A. Munoz-Villamizar, J. Santos, J. Montoya-Torres and M. Alvaréz,
Improving effectiveness of parallel machine scheduling with earliness and tardiness costs: A case study, International Journal of Industrial Engineering Computations, 10 (2019), 375-392.
doi: 10.5267/j.ijiec.2019.2.001. |
[27] |
J. Rezaeian, N. Derakhshan, I. Mahdavi and R. A. Foroutan,
Due date assignment and JIT scheduling problem in blocking hybrid flow shop robotic cells with multiple robots and batch delivery cost, International Journal of Industrial Mathematics, 13 (2021), 145-162.
|
[28] |
M. Savsar,
Simulation analysis of a pull-push system for an electronic assembly line, International Journal of Production Economics, 51 (1997), 205-214.
doi: 10.1016/S0925-5273(97)00055-8. |
[29] |
G. Taguchi, Introduction to quality engineering: Designing quality into products and processes, No. 658.562 T3, (1986). |
[30] |
E. Vallada and R. Ruiz,
A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times, European J. Oper. Res., 211 (2011), 612-622.
doi: 10.1016/j.ejor.2011.01.011. |
[31] |
B. Wang and H. Wang, Multiobjective order acceptance and scheduling on unrelated parallel machines with machine eligibility constraints, Math. Probl. Eng., 2018 (2018), 12pp.
doi: 10.1155/2018/6024631. |
[32] |
I. L. Wang, Y. C. Wang and C. W. Chen,
Scheduling unrelated parallel machines in semiconductor manufacturing by problem reduction and local search heuristics, Flexible Services and Manufacturing Journal, 25 (2013), 343-366.
doi: 10.1007/s10696-012-9150-7. |
[33] |
S. Wang, X. Wang, J. Yu, S. Ma and M. Liu,
Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan, Journal of Cleaner Production, 193 (2018), 424-440.
doi: 10.1016/j.jclepro.2018.05.056. |
[34] |
J. Xu, X. Xu and S. Q. Xie,
Recent developments in Dual Resource Constrained (DRC) system research, European Journal of Operational Research, 215 (2011), 309-318.
doi: 10.1016/j.ejor.2011.03.004. |
[35] |
K. C. Ying and S. W. Lin,
Unrelated parallel machine scheduling with sequence-and machine-dependent setup times and due date constraints, International Journal of Innovative Computing, 8 (2012), 3279-3297.
|
[36] |
A. Zhang, X. Qi and G. Li,
Machine scheduling with soft precedence constraints, European J. Oper. Res., 282 (2020), 491-505.
doi: 10.1016/j.ejor.2019.09.041. |
[37] |
J. R. Zeidi and S. MohammadHosseini,
Scheduling unrelated parallel machines with sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 81 (2015), 1487-1496.
doi: 10.1007/s00170-015-7215-y. |
[38] |
L. Zhang, Q. Deng, G. Gong and W. Han,
A new unrelated parallel machine scheduling problem with tool changes to minimise the total energy consumption, International Journal of Production Research, 58 (2020), 6826-6845.
doi: 10.1080/00207543.2019.1685708. |
[39] |
Z. Zhu and X. Zhou,
An efficient evolutionary grey wolf optimizer for multi-objective flexible job shop scheduling problem with hierarchical job precedence constraints, Computers & Industrial Engineering, 140 (2020), 106280.
doi: 10.1016/j.cie.2020.106280. |




















Constraints | |||||||
Reference | Unrelated Parallel machines | Worker flexibility/ allocation | Machine eligibility | Sequence dependent setup times | Precedence constraints | Objective function(s) | Solution approach(es) |
Cota et al. [7] | Yes | No | No | Yes | No | Makespan; Total energy consumption | ALNS, LA, MO-ALNS, MO-ALNS/D |
Zhu and Zhou [39] | No | No | No | No | Yes | Makespan; Total workload; Maximum machine workload | EMOGWO |
Munoz-Villamizar et al. [26] | No | No | No | Yes | No | Makespan; Total earliness and tardiness; Cost minimization; Effectiveness optimization | GAMS solver |
Arık and Toksarı [3] | No | No | No | No | No | Total tardiness penalty cost; Earliness penalty cost; Cost of setting due dates | A fuzzy local search algorithm |
Gong et al. [17] | No | Yes | No | No | No | Makespan; Total worker cost; Green production indicator | A hybrid evolutionary algorithm |
Zhang et al. [36] | No | No | No | No | Yes | Total penalties; Makespan; Total completion time | Approximation algorithms |
Gong et al. [16] | No | Yes | No | No | No | Makespan | A hybrid artificial bee colony |
Wang et al. [33] | No | No | No | No | No | Total energy consumption; Makespan | An augmented |
Zhang et al. [38] | Yes | No | No | No | No | Total energy consumption; Makespan | A heuristic evolutionary algorithm |
Lei et al. [24] | Yes | No | No | No | No | Makespan | An imperialist competitive algorithm |
Khanh Van and Van Hop [21] | Yes | No | No | Yes | No | Total weighted earliness and tardiness; Makespan | A hybrid algorithm based on GA and ISETP |
Kim et al. [22] | No | No | No | Yes | Yes | Total tardiness | SA, GA |
This paper | Yes | Yes | Yes | Yes | Yes | Cost of tardiness; Cost of hiring workers | MOTS, MOSA, A hybrid evolutionary algorithm |
Constraints | |||||||
Reference | Unrelated Parallel machines | Worker flexibility/ allocation | Machine eligibility | Sequence dependent setup times | Precedence constraints | Objective function(s) | Solution approach(es) |
Cota et al. [7] | Yes | No | No | Yes | No | Makespan; Total energy consumption | ALNS, LA, MO-ALNS, MO-ALNS/D |
Zhu and Zhou [39] | No | No | No | No | Yes | Makespan; Total workload; Maximum machine workload | EMOGWO |
Munoz-Villamizar et al. [26] | No | No | No | Yes | No | Makespan; Total earliness and tardiness; Cost minimization; Effectiveness optimization | GAMS solver |
Arık and Toksarı [3] | No | No | No | No | No | Total tardiness penalty cost; Earliness penalty cost; Cost of setting due dates | A fuzzy local search algorithm |
Gong et al. [17] | No | Yes | No | No | No | Makespan; Total worker cost; Green production indicator | A hybrid evolutionary algorithm |
Zhang et al. [36] | No | No | No | No | Yes | Total penalties; Makespan; Total completion time | Approximation algorithms |
Gong et al. [16] | No | Yes | No | No | No | Makespan | A hybrid artificial bee colony |
Wang et al. [33] | No | No | No | No | No | Total energy consumption; Makespan | An augmented |
Zhang et al. [38] | Yes | No | No | No | No | Total energy consumption; Makespan | A heuristic evolutionary algorithm |
Lei et al. [24] | Yes | No | No | No | No | Makespan | An imperialist competitive algorithm |
Khanh Van and Van Hop [21] | Yes | No | No | Yes | No | Total weighted earliness and tardiness; Makespan | A hybrid algorithm based on GA and ISETP |
Kim et al. [22] | No | No | No | Yes | Yes | Total tardiness | SA, GA |
This paper | Yes | Yes | Yes | Yes | Yes | Cost of tardiness; Cost of hiring workers | MOTS, MOSA, A hybrid evolutionary algorithm |
Job | ||||||
Job | ||||||
Test Problem | ||||||||||
with constraints | √ | √ | √ | √ | √ | √ | ||||
without constraints | √ | √ | √ | √ |
Test Problem | ||||||||||
with constraints | √ | √ | √ | √ | √ | √ | ||||
without constraints | √ | √ | √ | √ |
Parameter | Generating function |
Jobs processing time | |
Jobs delivery time | |
Setup times | |
Jobs tardiness penalties | |
Worker's skill |
Parameter | Generating function |
Jobs processing time | |
Jobs delivery time | |
Setup times | |
Jobs tardiness penalties | |
Worker's skill |
Algorithm | Parameter | Levels |
MOTS | ||
MOSA | ||
Hybrid | ||
Algorithm | Parameter | Levels |
MOTS | ||
MOSA | ||
Hybrid | ||
Algorithm | Parameter | Rank ( |
Rank (Means) | Best value |
MOTS | ||||
MOSA | ||||
Hybrid | ||||
Algorithm | Parameter | Rank ( |
Rank (Means) | Best value |
MOTS | ||||
MOSA | ||||
Hybrid | ||||
Problem |
Obj. functions | MOTS | MOSA | |||||
P1* | P2 | P3 | P4 | P1 | P2 | P3 | ||
f1 | 710 | 826 | 1661 | 1968 | ||||
f2 | 4853 | 4769 | 4769 | 4769 | ||||
f1 | 856 | 898 | 1550 | |||||
f2 | 3656 | 4418 | 4418 | |||||
f1 | 1063 | 1433 | ||||||
f2 | 3662 | 4880 | ||||||
f1 | 731 | 4762 | ||||||
f2 | 3065 | 3438 | ||||||
f1 | 983 | 1252 | 6937 | 6937 | ||||
f2 | 5719 | 5314 | 2055 | 2229 | ||||
f1 | 1460 | 3990 | ||||||
f2 | 6532 | 5869 | ||||||
f1 | 681 | 815 | 3837 | |||||
f2 | 6358 | 5880 | 5557 | |||||
f1 | 1207 | 1233 | 5487 | |||||
f2 | 6445 | 6184 | 5399 | |||||
f1 | 480 | 527 | 924 | 4346 | 4560 | |||
f2 | 10760 | 10542 | 10092 | 9807 | 9807 | |||
f1 | 2156 | 2176 | 2196 | 2579 | 5527 | |||
f2 | 9484 | 9224 | 8963 | 8703 | 9475 | |||
f1 | 6317 | 13816 | ||||||
f2 | 6943 | 9658 | ||||||
f1 | 2894 | 3466 | 3719 | 11392 | 12469 | |||
f2 | 8410 | 7044 | 6702 | 9826 | 9826 | |||
f1 | 12610 | 20565 | ||||||
f2 | 7749 | 12725 | ||||||
f1 | 3715 | 16032 | 16037 | |||||
f2 | 8823 | 11685 | 11685 | |||||
f1 | 4133 | 4560 | 19549 | 21010 | 22219 | |||
f2 | 14096 | 13584 | 15081 | 15081 | 15081 | |||
f1 | 3190 | 3444 | 3961 | 23057 | 25436 | |||
f2 | 13499 | 12942 | 12632 | 14132 | 14132 | |||
f1 | 8993 | 9944 | 38190 | 40236 | ||||
f2 | 17455 | 17182 | 21838 | 21838 | ||||
f1 | 6334 | 6421 | 7055 | 44881 | 31799 | |||
f2 | 18491 | 18383 | 18187 | 19731 | 19731 | |||
f1 | 13801 | 14655 | 73855 | 78444 | 80185 | |||
f2 | 19918 | 19450 | 19905 | 19905 | 19905 | |||
f1 | 11484 | 13550 | 47375 | 51987 | 59331 | |||
f2 | 22310 | 21220 | 26906 | 26906 | 26906 | |||
Problem |
Obj. functions | MOTS | MOSA | |||||
P1* | P2 | P3 | P4 | P1 | P2 | P3 | ||
f1 | 710 | 826 | 1661 | 1968 | ||||
f2 | 4853 | 4769 | 4769 | 4769 | ||||
f1 | 856 | 898 | 1550 | |||||
f2 | 3656 | 4418 | 4418 | |||||
f1 | 1063 | 1433 | ||||||
f2 | 3662 | 4880 | ||||||
f1 | 731 | 4762 | ||||||
f2 | 3065 | 3438 | ||||||
f1 | 983 | 1252 | 6937 | 6937 | ||||
f2 | 5719 | 5314 | 2055 | 2229 | ||||
f1 | 1460 | 3990 | ||||||
f2 | 6532 | 5869 | ||||||
f1 | 681 | 815 | 3837 | |||||
f2 | 6358 | 5880 | 5557 | |||||
f1 | 1207 | 1233 | 5487 | |||||
f2 | 6445 | 6184 | 5399 | |||||
f1 | 480 | 527 | 924 | 4346 | 4560 | |||
f2 | 10760 | 10542 | 10092 | 9807 | 9807 | |||
f1 | 2156 | 2176 | 2196 | 2579 | 5527 | |||
f2 | 9484 | 9224 | 8963 | 8703 | 9475 | |||
f1 | 6317 | 13816 | ||||||
f2 | 6943 | 9658 | ||||||
f1 | 2894 | 3466 | 3719 | 11392 | 12469 | |||
f2 | 8410 | 7044 | 6702 | 9826 | 9826 | |||
f1 | 12610 | 20565 | ||||||
f2 | 7749 | 12725 | ||||||
f1 | 3715 | 16032 | 16037 | |||||
f2 | 8823 | 11685 | 11685 | |||||
f1 | 4133 | 4560 | 19549 | 21010 | 22219 | |||
f2 | 14096 | 13584 | 15081 | 15081 | 15081 | |||
f1 | 3190 | 3444 | 3961 | 23057 | 25436 | |||
f2 | 13499 | 12942 | 12632 | 14132 | 14132 | |||
f1 | 8993 | 9944 | 38190 | 40236 | ||||
f2 | 17455 | 17182 | 21838 | 21838 | ||||
f1 | 6334 | 6421 | 7055 | 44881 | 31799 | |||
f2 | 18491 | 18383 | 18187 | 19731 | 19731 | |||
f1 | 13801 | 14655 | 73855 | 78444 | 80185 | |||
f2 | 19918 | 19450 | 19905 | 19905 | 19905 | |||
f1 | 11484 | 13550 | 47375 | 51987 | 59331 | |||
f2 | 22310 | 21220 | 26906 | 26906 | 26906 | |||
Problem |
Obj. functions | Hybrid | ||||||||||
P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | P10 | P11 | ||
f1 | 1524 | 1553 | 1617 | |||||||||
f2 | 5191 | 5107 | 5022 | |||||||||
f1 | 1216 | 1400 | 1624 | 1961 | ||||||||
f2 | 4266 | 4266 | 4113 | 3961 | ||||||||
f1 | 989 | 1405 | 1677 | 1711 | 1791 | 1867 | 1870 | 1896 | ||||
f2 | 6097 | 5793 | 5793 | 5488 | 5184 | 5184 | 4880 | 4575 | ||||
f1 | 2443 | 2586 | 2671 | 2695 | 2702 | 2704 | ||||||
f2 | 3295 | 3295 | 3208 | 3152 | 3152 | 3065 | ||||||
f1 | 858 | 1047 | 1196 | 1330 | 1452 | 1601 | ||||||
f2 | 6531 | 6531 | 6328 | 6125 | 5922 | 5719 | ||||||
f1 | 2106 | 2400 | 2491 | 2841 | 2946 | |||||||
f2 | 7526 | 7195 | 6863 | 6532 | 6201 | |||||||
f1 | 1428 | 2331 | 2669 | 2690 | 3088 | 3109 | 3258 | 3279 | 3334 | 3438 | ||
f2 | 8268 | 7790 | 7313 | 7046 | 6835 | 6568 | 6358 | 6091 | 5880 | 5613 | ||
f1 | 3812 | 3829 | ||||||||||
f2 | 6707 | 6184 | ||||||||||
f1 | 2871 | 3015 | 3589 | 3610 | 3617 | 3618 | 3619 | |||||
f2 | 1080 | 1080 | 1078 | 1077 | 1074 | 1054 | 1052 | |||||
f1 | 2100 | 2313 | 2576 | 2600 | ||||||||
f2 | 9739 | 9739 | 9484 | 9483 | ||||||||
f1 | 6887 | 6979 | 7519 | |||||||||
f2 | 11292 | 10078 | 9755 | |||||||||
f1 | 5712 | 7395 | 7406 | 7445 | 9537 | 9551 | 9718 | 9740 | 10162 | 10176 | 10365 | |
f2 | 9718 | 9718 | 9587 | 9518 | 9518 | 9386 | 9309 | 9253 | 9044 | 8782 | 8779 | |
f1 | 11769 | 11826 | 12126 | 13470 | 17244 | 17665 | ||||||
f2 | 10769 | 10769 | 10598 | 10598 | 10598 | 10237 | ||||||
f1 | 5851 | 6530 | 6679 | 6684 | 8501 | 8534 | 8715 | 9107 | 9122 | |||
f2 | 10561 | 10561 | 10389 | 10266 | 10240 | 10117 | 10114 | 10114 | 9991 | |||
f1 | 13183 | 13805 | 13942 | 14068 | 15660 | 16033 | 17741 | 17785 | ||||
f2 | 16567 | 16411 | 16391 | 16211 | 16191 | 16010 | 15498 | 15142 | ||||
f1 | 10899 | 10951 | 10968 | 13811 | 17298 | 18530 | 18552 | |||||
f2 | 15173 | 15031 | 14762 | 14762 | 14566 | 14084 | 13899 | |||||
f1 | 21130 | 22146 | 23254 | |||||||||
f2 | 25930 | 25229 | 24604 | |||||||||
f1 | 18544 | 18551 | 20567 | 20619 | 20629 | 21045 | ||||||
f2 | 21022 | 20914 | 20914 | 20910 | 20866 | 20866 | ||||||
f1 | 47230 | 47348 | 60202 | 62672 | ||||||||
f2 | 23776 | 23729 | 23456 | 21764 | ||||||||
f1 | 18791 | 23480 | 26830 | 28710 | 28254 | 29460 | ||||||
f2 | 28995 | 28995 | 28637 | 28376 | 28302 | 28202 |
Problem |
Obj. functions | Hybrid | ||||||||||
P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | P10 | P11 | ||
f1 | 1524 | 1553 | 1617 | |||||||||
f2 | 5191 | 5107 | 5022 | |||||||||
f1 | 1216 | 1400 | 1624 | 1961 | ||||||||
f2 | 4266 | 4266 | 4113 | 3961 | ||||||||
f1 | 989 | 1405 | 1677 | 1711 | 1791 | 1867 | 1870 | 1896 | ||||
f2 | 6097 | 5793 | 5793 | 5488 | 5184 | 5184 | 4880 | 4575 | ||||
f1 | 2443 | 2586 | 2671 | 2695 | 2702 | 2704 | ||||||
f2 | 3295 | 3295 | 3208 | 3152 | 3152 | 3065 | ||||||
f1 | 858 | 1047 | 1196 | 1330 | 1452 | 1601 | ||||||
f2 | 6531 | 6531 | 6328 | 6125 | 5922 | 5719 | ||||||
f1 | 2106 | 2400 | 2491 | 2841 | 2946 | |||||||
f2 | 7526 | 7195 | 6863 | 6532 | 6201 | |||||||
f1 | 1428 | 2331 | 2669 | 2690 | 3088 | 3109 | 3258 | 3279 | 3334 | 3438 | ||
f2 | 8268 | 7790 | 7313 | 7046 | 6835 | 6568 | 6358 | 6091 | 5880 | 5613 | ||
f1 | 3812 | 3829 | ||||||||||
f2 | 6707 | 6184 | ||||||||||
f1 | 2871 | 3015 | 3589 | 3610 | 3617 | 3618 | 3619 | |||||
f2 | 1080 | 1080 | 1078 | 1077 | 1074 | 1054 | 1052 | |||||
f1 | 2100 | 2313 | 2576 | 2600 | ||||||||
f2 | 9739 | 9739 | 9484 | 9483 | ||||||||
f1 | 6887 | 6979 | 7519 | |||||||||
f2 | 11292 | 10078 | 9755 | |||||||||
f1 | 5712 | 7395 | 7406 | 7445 | 9537 | 9551 | 9718 | 9740 | 10162 | 10176 | 10365 | |
f2 | 9718 | 9718 | 9587 | 9518 | 9518 | 9386 | 9309 | 9253 | 9044 | 8782 | 8779 | |
f1 | 11769 | 11826 | 12126 | 13470 | 17244 | 17665 | ||||||
f2 | 10769 | 10769 | 10598 | 10598 | 10598 | 10237 | ||||||
f1 | 5851 | 6530 | 6679 | 6684 | 8501 | 8534 | 8715 | 9107 | 9122 | |||
f2 | 10561 | 10561 | 10389 | 10266 | 10240 | 10117 | 10114 | 10114 | 9991 | |||
f1 | 13183 | 13805 | 13942 | 14068 | 15660 | 16033 | 17741 | 17785 | ||||
f2 | 16567 | 16411 | 16391 | 16211 | 16191 | 16010 | 15498 | 15142 | ||||
f1 | 10899 | 10951 | 10968 | 13811 | 17298 | 18530 | 18552 | |||||
f2 | 15173 | 15031 | 14762 | 14762 | 14566 | 14084 | 13899 | |||||
f1 | 21130 | 22146 | 23254 | |||||||||
f2 | 25930 | 25229 | 24604 | |||||||||
f1 | 18544 | 18551 | 20567 | 20619 | 20629 | 21045 | ||||||
f2 | 21022 | 20914 | 20914 | 20910 | 20866 | 20866 | ||||||
f1 | 47230 | 47348 | 60202 | 62672 | ||||||||
f2 | 23776 | 23729 | 23456 | 21764 | ||||||||
f1 | 18791 | 23480 | 26830 | 28710 | 28254 | 29460 | ||||||
f2 | 28995 | 28995 | 28637 | 28376 | 28302 | 28202 |
Metric | Problem size | Sum of Squares | df | Mean Square | F. | Sig. |
S-metric | Small | 8758922.903 | 2 | 4379461.452 | 5.519 | 0.012 |
Medium | 8.541E7 | 2 | 4.271E7 | 3.688 | 0.050 | |
Large | 1.348E9 | 2 | 6.739E8 | 6.716 | 0.008 | |
NP | Small | 130.083 | 2 | 65.042 | 24.500 | 0.000 |
Medium | 81.333 | 2 | 40.667 | 7.562 | 0.005 | |
Large | 42.333 | 2 | 21.167 | 15.744 | 0.000 | |
MID | Small | 6748717.583 | 2 | 3374358.792 | 2.312 | 0.124 |
Medium | 1.932E8 | 2 | 9.662E7 | 4.538 | 0.029 | |
Large | 2.856E9 | 2 | 1.428E9 | 11.064 | 0.001 | |
DM | Small | 8430960.250 | 2 | 4215480.125 | 10.568 | 0.001 |
Medium | 7.217E7 | 2 | 3.608E7 | 7.737 | 0.005 | |
Large | 1.954E8 | 2 | 9.768E7 | 5.840 | 0.013 | |
SNS | Small | 122196.583 | 2 | 61098.292 | 5.399 | 0.013 |
Medium | 1.564E7 | 2 | 7817526.167 | 2.349 | 0.130 | |
Large | 1.236E7 | 2 | 6179340.222 | 0.094 | 0.911 | |
Time | Small | 20371.750 | 2 | 10185.875 | 21.221 | 0.000 |
Medium | 43694.111 | 2 | 21847.056 | 6.425 | 0.010 | |
Large | 177652.111 | 2 | 88826.056 | 10.736 | 0.001 |
Metric | Problem size | Sum of Squares | df | Mean Square | F. | Sig. |
S-metric | Small | 8758922.903 | 2 | 4379461.452 | 5.519 | 0.012 |
Medium | 8.541E7 | 2 | 4.271E7 | 3.688 | 0.050 | |
Large | 1.348E9 | 2 | 6.739E8 | 6.716 | 0.008 | |
NP | Small | 130.083 | 2 | 65.042 | 24.500 | 0.000 |
Medium | 81.333 | 2 | 40.667 | 7.562 | 0.005 | |
Large | 42.333 | 2 | 21.167 | 15.744 | 0.000 | |
MID | Small | 6748717.583 | 2 | 3374358.792 | 2.312 | 0.124 |
Medium | 1.932E8 | 2 | 9.662E7 | 4.538 | 0.029 | |
Large | 2.856E9 | 2 | 1.428E9 | 11.064 | 0.001 | |
DM | Small | 8430960.250 | 2 | 4215480.125 | 10.568 | 0.001 |
Medium | 7.217E7 | 2 | 3.608E7 | 7.737 | 0.005 | |
Large | 1.954E8 | 2 | 9.768E7 | 5.840 | 0.013 | |
SNS | Small | 122196.583 | 2 | 61098.292 | 5.399 | 0.013 |
Medium | 1.564E7 | 2 | 7817526.167 | 2.349 | 0.130 | |
Large | 1.236E7 | 2 | 6179340.222 | 0.094 | 0.911 | |
Time | Small | 20371.750 | 2 | 10185.875 | 21.221 | 0.000 |
Medium | 43694.111 | 2 | 21847.056 | 6.425 | 0.010 | |
Large | 177652.111 | 2 | 88826.056 | 10.736 | 0.001 |
Small | Medium | Large | ||||
Metrics | Order of best performances | Significant difference | Order of best performances | Significant difference | Order of best performances | Significant difference |
S-metric | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA |
NPS | Hybrid-MOTS-MOSA | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS |
MID | Hybrid-MOTS-MOSA | - | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA |
DM | Hybrid-MOTS-MOSA | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | Hybrid-MOTS |
SNS | Hybrid-MOTS-MOSA | Hybrid-MOSA | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | - |
Time | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOSA-MOTS | Hybrid-MOTS | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS |
Small | Medium | Large | ||||
Metrics | Order of best performances | Significant difference | Order of best performances | Significant difference | Order of best performances | Significant difference |
S-metric | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA |
NPS | Hybrid-MOTS-MOSA | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS |
MID | Hybrid-MOTS-MOSA | - | MOTS-Hybrid-MOSA | MOTS-MOSA | MOTS-Hybrid-MOSA | MOTS-MOSA |
DM | Hybrid-MOTS-MOSA | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | Hybrid-MOTS |
SNS | Hybrid-MOTS-MOSA | Hybrid-MOSA | Hybrid-MOTS-MOSA | - | Hybrid-MOSA-MOTS | - |
Time | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS | Hybrid-MOSA-MOTS | Hybrid-MOTS | Hybrid-MOSA-MOTS | Hybrid-MOSA, Hybrid-MOTS |
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