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Dynamic virtual cellular reconfiguration for capacity planning of market-oriented production systems
| 1. | Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan city, Hubei province, China |
| 2. | School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou city, China |
Market-oriented production systems generally have the characteristics of multi-product and small-batch production. Dynamic virtual cellular manufacturing systems create virtual manufacturing cells periodically in a planning horizon to respond to changing demands flexibly and quickly, and thus are suitable for production planning problems of market-oriented production systems. In the current research, we propose a dynamic virtual cell reconfiguration framework under a dynamic environment with unstable demands and multiple planning cycles. In this framework, we formulate a two-phase dynamic virtual cell formation (DVCF) model. In the first phase, the proposed model aims to maximize processing similarity and balance the workload in the system. In the second phase, we consider the objective of reconfiguration stability based on the first phase model. To address the proposed model, we design a hybrid metaheuristic named Lévy-NSGA-Ⅱ, and perform various computational experiments to examine the performance of the proposed algorithm. Results of experiments indicate that the proposed Lévy-NSGA-Ⅱ based algorithm outperforms multi-objective cuckoo search (MOCS) and NSGA-Ⅱ in solution quality and optimal solution size.
Keywords:
Dynamic virtual cell formation,
multi-objective optimization,
hybrid metaheuristic,
discrete Lévy flight search strategy,
capacity planning.
Mathematics Subject Classification:
Primary: 90B50, 90B30.
Citation:
Zhengmin Zhang, Zailin Guan, Weikang Fang, Lei Yue.
Dynamic virtual cellular reconfiguration for capacity planning of market-oriented production systems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022009
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Solving the integrated cell formation and worker assignment problem using particle swarm optimization and linear programming, Computers & Industrial Engineering, 110 (2017), 126-137.
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Virtual cellular inheritance reconfiguration driven by random arrival orders and time window, Jisuanji Jicheng Zhizao Xitong/Computer Integrated Manufacturing Systems, CIMS, 24 (2018), 1317-1326.
|
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A. Hosseini, M. M. Paydar, I. Mahdavi and J. Jouzdani,
Cell forming and cell balancing of virtual cellular manufacturing systems with alternative processing routes using genetic algorithm, Journal of Optimization in Industrial Engineering, 9 (2016), 41-51.
|
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M. Imran, C. Kang, Y. H. Lee, M. Jahanzaib and H. Aziz,
Cell formation in a cellular manufacturing system using simulation integrated hybrid genetic algorithm, Computers & Industrial Engineering, 105 (2017), 123-135.
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Optimization of the material handling costs and the machine reliability in cellular manufacturing system using cuckoo search algorithm, Neural Computing and Applications, 31 (2019), 3743-3757.
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R. Kia, N. Javadian, M. M. Paydar and M. Saidi-Mehrabad, A simulated annealing for intra-cell layout design of dynamic cellular manufacturing systems with route selection, purchasing machines and cell reconfiguration, Asia-Pac. J. Oper. Res., 30 (2013), 1350004, 41 pp.
doi: 10.1142/S0217595913500048. |
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A. Kusiak,
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J. Li, A. Wang and C. Tang,
Production planning in virtual cell of reconfiguration manufacturing system using genetic algorithm, The International Journal of Advanced Manufacturing Technology, 74 (2014), 47-64.
doi: 10.1007/s00170-014-5987-0. |
| [21] |
J. Q. Li, Q. K. Pan and M. F. Tasgetiren,
A discrete artificial bee colony algorithm for the multi-objective flexible job-shop scheduling problem with maintenance activities, Appl. Math. Model., 38 (2014), 1111-1132.
doi: 10.1016/j.apm.2013.07.038. |
| [22] |
J. Q. Li, Q. K. Pan and F. T. Wang,
A hybrid variable neighborhood search for solving the hybrid flow shop scheduling problem, Applied Soft Computing, 24 (2014), 63-77.
doi: 10.1016/j.asoc.2014.07.005. |
| [23] |
J. Q. Li, H. Y. Sang, Y. Y. Han, C. G. Wang and K. Z. Gao,
Efficient multi-objective optimization algorithm for hybrid flow shop scheduling problems with setup energy consumptions, Journal of Cleaner Production, 181 (2018), 584-598.
doi: 10.1016/j.jclepro.2018.02.004. |
| [24] |
C. Liu, J. Wang and M. Zhou,
Reconfiguration of virtual cellular manufacturing systems via improved imperialist competitive approach, IEEE Transactions on Automation Science and Engineering, 16 (2018), 1301-1314.
doi: 10.1109/TASE.2018.2878653. |
| [25] |
I. Mahdavi, A. Aalaei, M. M. Paydar and M. Solimanpur,
Multi-objective cell formation and production planning in dynamic virtual cellular manufacturing systems, International Journal of Production Research, 49 (2011), 6517-6537.
doi: 10.1080/00207543.2010.524902. |
| [26] |
P. M. Mahdi and S.-M. Mohammad,
A hybrid genetic algorithm for dynamic virtual cellular manufacturing with supplier selection, The International Journal of Advanced Manufacturing Technology, 92 (2017), 3001-3017.
|
| [27] |
M. Mohammadi and K. Forghani,
A hybrid method based on genetic algorithm and dynamic programming for solving a bi-objective cell formation problem considering alternative process routings and machine duplication, Applied Soft Computing, 53 (2017), 97-110.
doi: 10.1016/j.asoc.2016.12.039. |
| [28] |
M. Moradgholi, M. M. Paydar, I. Mahdavi and J. Jouzdani,
A genetic algorithm for a bi-objective mathematical model for dynamic virtual cell formation problem, Journal of Industrial Engineering International, 12 (2016), 343-359.
doi: 10.1007/s40092-016-0151-0. |
| [29] |
J. S. Morris and R. J. Tersine,
A simulation analysis of factors influencing the attractiveness of group technology cellular layouts, Management Science, 36 (1990), 1567-1578.
|
| [30] |
F. Niakan, A. Baboli, T. Moyaux and V. Botta-Genoulaz,
A bi-objective model in sustainable dynamic cell formation problem with skill-based worker assignment, Journal of Manufacturing Systems, 38 (2016), 46-62.
doi: 10.1016/j.jmsy.2015.11.001. |
| [31] |
E. Nikoofarid and A. Aalaei,
Production planning and worker assignment in a dynamic virtual cellular manufacturing system, International Journal of Management Science and Engineering Management, 7 (2012), 89-95.
doi: 10.1080/17509653.2012.10671211. |
| [32] |
Q. K. Pan, L. Wang and B. Qian,
A novel differential evolution algorithm for bi-criteria no-wait flow shop scheduling problems, Comput. Oper. Res., 36 (2009), 2498-2511.
doi: 10.1016/j.cor.2008.10.008. |
| [33] |
M. Rabbani, H. Farrokhi-Asl and M. Ravanbakhsh,
Dynamic cellular manufacturing system considering machine failure and workload balance, Journal of Industrial Engineering International, 15 (2019), 25-40.
doi: 10.1007/s40092-018-0261-y. |
| [34] |
V. Rahimi, J. Arkat and H. Farughi,
A vibration damping optimization algorithm for the integrated problem of cell formation, cellular scheduling, and intercellular layout, Computers & Industrial Engineering, 143 (2020), 106439.
doi: 10.1016/j.cie.2020.106439. |
| [35] |
J. Rezaeian, N. Javadian, R. Tavakkoli-Moghaddam and F. Jolai,
A hybrid approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system, Applied Soft Computing, 11 (2011), 4195-4202.
doi: 10.1016/j.asoc.2011.03.013. |
| [36] |
D. Rogers and S. Shafer, Measuring cellular manufacturing performance, In Manufacturing Research and Technology, 24 1995,147–165.
doi: 10.1016/S1572-4417(06)80040-9. |
| [37] |
B. Sarker,
Grouping efficiency measures in cellular manufacturing: A survey and critical review, International Journal of Production Research, 37 (1999), 285-314.
doi: 10.1080/002075499191779. |
| [38] |
G. Syswerda, Uniform crossover in genetic algorithms, In Proceedings of the Third International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, (1989), 2–9. |
| [39] |
G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. Da Luz, E. Raposo and H. E. Stanley,
Optimizing the success of random searches, Nature, 401 (1999), 911-914.
doi: 10.1038/44831. |
| [40] |
G. Xue and O. F. Offodile,
Integrated optimization of dynamic cell formation and hierarchical production planning problems, Computers & Industrial Engineering, 139 (2020), 106155.
doi: 10.1016/j.cie.2019.106155. |
| [41] |
X. S. Yang and S. Deb, Cuckoo search via lévy flights, In 2009 World Congress on Nature & Biologically inspired computing (NaBIC), Ieee, (2009), 210–214. |
show all references
References:
| [1] |
T. Blickle and L. Thiele,
A comparison of selection schemes used in evolutionary algorithms, Evolutionary Computation, 4 (1996), 361-394.
doi: 10.1162/evco.1996.4.4.361. |
| [2] |
M. Bortolini, E. Ferrari, F. G. Galizia and A. Regattieri,
An optimisation model for the dynamic management of cellular reconfigurable manufacturing systems under auxiliary module availability constraints, Journal of Manufacturing Systems, 58 (2021), 442-451.
doi: 10.1016/j.jmsy.2021.01.001. |
| [3] |
C. W. Chou, C. F. Chien and M. Gen,
A multiobjective hybrid genetic algorithm for tft-lcd module assembly scheduling, IEEE Transactions on Automation Science and Engineering, 11 (2014), 692-705.
doi: 10.1109/TASE.2014.2316193. |
| [4] |
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan,
A fast and elitist multiobjective genetic algorithm: Nsga-ii, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.
doi: 10.1109/4235.996017. |
| [5] |
K. Deep and P. K. Singh,
Design of robust cellular manufacturing system for dynamic part population considering multiple processing routes using genetic algorithm, Journal of Manufacturing Systems, 35 (2015), 155-163.
doi: 10.1016/j.jmsy.2014.09.008. |
| [6] |
I. Eguia, J. C. Molina, S. Lozano and J. Racero,
Cell design and multi-period machine loading in cellular reconfigurable manufacturing systems with alternative routing, International Journal of Production Research, 55 (2017), 2775-2790.
doi: 10.1080/00207543.2016.1193673. |
| [7] |
B. Erfani, S. Ebrahimnejad and A. Moosavi,
An integrated dynamic facility layout and job shop scheduling problem: A hybrid nsga-ii and local search algorithm, J. Ind. Manag. Optim., 16 (2020), 1801-1834.
doi: 10.3934/jimo.2019030. |
| [8] |
J. Fan and D. Feng,
Design of cellular manufacturing system with quasi-dynamic dual resource using multi-objective ga, International Journal of Production Research, 51 (2013), 4134-4154.
doi: 10.1080/00207543.2012.748228. |
| [9] |
H. Feng, W. Da, L. Xi, E. Pan and T. Xia,
Solving the integrated cell formation and worker assignment problem using particle swarm optimization and linear programming, Computers & Industrial Engineering, 110 (2017), 126-137.
doi: 10.1016/j.cie.2017.05.038. |
| [10] |
R. Y. Fung, F. Liang, Z. Jiang and T. Wong,
A multi-stage methodology for virtual cell formation oriented agile manufacturing, The International Journal of Advanced Manufacturing Technology, 36 (2008), 798-810.
doi: 10.1007/s00170-006-0871-1. |
| [11] |
H. Guo, M. Chen, K. Mohamed, T. Qu, S. Wang and J. Li,
A digital twin-based flexible cellular manufacturing for optimization of air conditioner line, Journal of Manufacturing Systems, 58 (2021), 65-78.
doi: 10.1016/j.jmsy.2020.07.012. |
| [12] |
W. Hachicha, F. Masmoudi and M. Haddar,
Formation of machine groups and part families in cellular manufacturing systems using a correlation analysis approach, The International Journal of Advanced Manufacturing Technology, 36 (2008), 1157-1169.
doi: 10.1007/s00170-007-0928-9. |
| [13] |
M. Hamedi, G. Esmaeilian, N. Ismail and M. Ariffin,
A survey on formation of virtual cellular manufacturing systems (vcmss) and related issues, Scientific Research and Essays, 7 (2012), 3316-3328.
|
| [14] |
W. Han, Y. Yu, L. Gao, J. Fang and Z. Li,
Virtual cellular inheritance reconfiguration driven by random arrival orders and time window, Jisuanji Jicheng Zhizao Xitong/Computer Integrated Manufacturing Systems, CIMS, 24 (2018), 1317-1326.
|
| [15] |
A. Hosseini, M. M. Paydar, I. Mahdavi and J. Jouzdani,
Cell forming and cell balancing of virtual cellular manufacturing systems with alternative processing routes using genetic algorithm, Journal of Optimization in Industrial Engineering, 9 (2016), 41-51.
|
| [16] |
M. Imran, C. Kang, Y. H. Lee, M. Jahanzaib and H. Aziz,
Cell formation in a cellular manufacturing system using simulation integrated hybrid genetic algorithm, Computers & Industrial Engineering, 105 (2017), 123-135.
doi: 10.1016/j.cie.2016.12.028. |
| [17] |
B. Karoum and Y. B. Elbenani,
Optimization of the material handling costs and the machine reliability in cellular manufacturing system using cuckoo search algorithm, Neural Computing and Applications, 31 (2019), 3743-3757.
doi: 10.1007/s00521-017-3302-3. |
| [18] |
R. Kia, N. Javadian, M. M. Paydar and M. Saidi-Mehrabad, A simulated annealing for intra-cell layout design of dynamic cellular manufacturing systems with route selection, purchasing machines and cell reconfiguration, Asia-Pac. J. Oper. Res., 30 (2013), 1350004, 41 pp.
doi: 10.1142/S0217595913500048. |
| [19] |
A. Kusiak,
The generalized group technology concept, International journal of production research, 25 (1987), 561-569.
doi: 10.1080/00207548708919861. |
| [20] |
J. Li, A. Wang and C. Tang,
Production planning in virtual cell of reconfiguration manufacturing system using genetic algorithm, The International Journal of Advanced Manufacturing Technology, 74 (2014), 47-64.
doi: 10.1007/s00170-014-5987-0. |
| [21] |
J. Q. Li, Q. K. Pan and M. F. Tasgetiren,
A discrete artificial bee colony algorithm for the multi-objective flexible job-shop scheduling problem with maintenance activities, Appl. Math. Model., 38 (2014), 1111-1132.
doi: 10.1016/j.apm.2013.07.038. |
| [22] |
J. Q. Li, Q. K. Pan and F. T. Wang,
A hybrid variable neighborhood search for solving the hybrid flow shop scheduling problem, Applied Soft Computing, 24 (2014), 63-77.
doi: 10.1016/j.asoc.2014.07.005. |
| [23] |
J. Q. Li, H. Y. Sang, Y. Y. Han, C. G. Wang and K. Z. Gao,
Efficient multi-objective optimization algorithm for hybrid flow shop scheduling problems with setup energy consumptions, Journal of Cleaner Production, 181 (2018), 584-598.
doi: 10.1016/j.jclepro.2018.02.004. |
| [24] |
C. Liu, J. Wang and M. Zhou,
Reconfiguration of virtual cellular manufacturing systems via improved imperialist competitive approach, IEEE Transactions on Automation Science and Engineering, 16 (2018), 1301-1314.
doi: 10.1109/TASE.2018.2878653. |
| [25] |
I. Mahdavi, A. Aalaei, M. M. Paydar and M. Solimanpur,
Multi-objective cell formation and production planning in dynamic virtual cellular manufacturing systems, International Journal of Production Research, 49 (2011), 6517-6537.
doi: 10.1080/00207543.2010.524902. |
| [26] |
P. M. Mahdi and S.-M. Mohammad,
A hybrid genetic algorithm for dynamic virtual cellular manufacturing with supplier selection, The International Journal of Advanced Manufacturing Technology, 92 (2017), 3001-3017.
|
| [27] |
M. Mohammadi and K. Forghani,
A hybrid method based on genetic algorithm and dynamic programming for solving a bi-objective cell formation problem considering alternative process routings and machine duplication, Applied Soft Computing, 53 (2017), 97-110.
doi: 10.1016/j.asoc.2016.12.039. |
| [28] |
M. Moradgholi, M. M. Paydar, I. Mahdavi and J. Jouzdani,
A genetic algorithm for a bi-objective mathematical model for dynamic virtual cell formation problem, Journal of Industrial Engineering International, 12 (2016), 343-359.
doi: 10.1007/s40092-016-0151-0. |
| [29] |
J. S. Morris and R. J. Tersine,
A simulation analysis of factors influencing the attractiveness of group technology cellular layouts, Management Science, 36 (1990), 1567-1578.
|
| [30] |
F. Niakan, A. Baboli, T. Moyaux and V. Botta-Genoulaz,
A bi-objective model in sustainable dynamic cell formation problem with skill-based worker assignment, Journal of Manufacturing Systems, 38 (2016), 46-62.
doi: 10.1016/j.jmsy.2015.11.001. |
| [31] |
E. Nikoofarid and A. Aalaei,
Production planning and worker assignment in a dynamic virtual cellular manufacturing system, International Journal of Management Science and Engineering Management, 7 (2012), 89-95.
doi: 10.1080/17509653.2012.10671211. |
| [32] |
Q. K. Pan, L. Wang and B. Qian,
A novel differential evolution algorithm for bi-criteria no-wait flow shop scheduling problems, Comput. Oper. Res., 36 (2009), 2498-2511.
doi: 10.1016/j.cor.2008.10.008. |
| [33] |
M. Rabbani, H. Farrokhi-Asl and M. Ravanbakhsh,
Dynamic cellular manufacturing system considering machine failure and workload balance, Journal of Industrial Engineering International, 15 (2019), 25-40.
doi: 10.1007/s40092-018-0261-y. |
| [34] |
V. Rahimi, J. Arkat and H. Farughi,
A vibration damping optimization algorithm for the integrated problem of cell formation, cellular scheduling, and intercellular layout, Computers & Industrial Engineering, 143 (2020), 106439.
doi: 10.1016/j.cie.2020.106439. |
| [35] |
J. Rezaeian, N. Javadian, R. Tavakkoli-Moghaddam and F. Jolai,
A hybrid approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system, Applied Soft Computing, 11 (2011), 4195-4202.
doi: 10.1016/j.asoc.2011.03.013. |
| [36] |
D. Rogers and S. Shafer, Measuring cellular manufacturing performance, In Manufacturing Research and Technology, 24 1995,147–165.
doi: 10.1016/S1572-4417(06)80040-9. |
| [37] |
B. Sarker,
Grouping efficiency measures in cellular manufacturing: A survey and critical review, International Journal of Production Research, 37 (1999), 285-314.
doi: 10.1080/002075499191779. |
| [38] |
G. Syswerda, Uniform crossover in genetic algorithms, In Proceedings of the Third International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, (1989), 2–9. |
| [39] |
G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. Da Luz, E. Raposo and H. E. Stanley,
Optimizing the success of random searches, Nature, 401 (1999), 911-914.
doi: 10.1038/44831. |
| [40] |
G. Xue and O. F. Offodile,
Integrated optimization of dynamic cell formation and hierarchical production planning problems, Computers & Industrial Engineering, 139 (2020), 106155.
doi: 10.1016/j.cie.2019.106155. |
| [41] |
X. S. Yang and S. Deb, Cuckoo search via lévy flights, In 2009 World Congress on Nature & Biologically inspired computing (NaBIC), Ieee, (2009), 210–214. |
Figure 2.
Discrete Lévy flight search strategy
Figure 3.
Flow chart of Lévy-NSGA-Ⅱ
Figure 4.
Schematic structure of the chromosomes in Lévy-NSGA-II
Figure 5.
Number of machine allocation for all cells
Figure 6.
Crossover and mutation operations
Figure 7.
Local random search
Figure 8.
Global random search
Figure 9.
The applications of Lévy-NSGA-II in the two-phase DVCF model
Figure 10.
Pareto-optimal solutions for A1-A6
Figure 11.
Pareto-optimal solutions for B1-B6
Figure 12.
Comparisons of Lévy-NSGA-II, NSGA-II, and MOCS
Figure 13.
The Pareto-optimal solutions for the first phase
Figure 14.
The Pareto-optimal solutions for the second period
Table 1.
Notations used in the DVCF model
| Indices | |
| part types, |
|
| process routings, |
|
| machine types, |
|
| virtual cells, |
|
| formation periods | |
| Input parameters | |
| number of part types in period |
|
| number of process routings for part type |
|
| M | number of machine types |
| number of machines included in machine type |
|
| upper bounds of virtual cells | |
| lower bounds of virtual cells | |
| demand for part type |
|
| production capacity of each machine of type |
|
| 1, if |
|
| processing time of |
|
| the similarity coefficient between |
|
| number of machines of type |
|
| Variables | |
| number of virtual cells in period |
|
| 1, part type |
|
| number of machines of type |
|
| number of machines of type |
|
| Indices | |
| part types, |
|
| process routings, |
|
| machine types, |
|
| virtual cells, |
|
| formation periods | |
| Input parameters | |
| number of part types in period |
|
| number of process routings for part type |
|
| M | number of machine types |
| number of machines included in machine type |
|
| upper bounds of virtual cells | |
| lower bounds of virtual cells | |
| demand for part type |
|
| production capacity of each machine of type |
|
| 1, if |
|
| processing time of |
|
| the similarity coefficient between |
|
| number of machines of type |
|
| Variables | |
| number of virtual cells in period |
|
| 1, part type |
|
| number of machines of type |
|
| number of machines of type |
|
Table 2.
Pattern of data generation
| Parameter | Generation pattern | Parameter | Generation pattern |
| 8 | U [1,3] | ||
| U [2J, 3J] | U [2,6] | ||
| Random{3, 4} | U 10 * [4,32] | ||
| Random{2, 3} | U 100 * [10,30] | ||
| U [0, 10] |
| Parameter | Generation pattern | Parameter | Generation pattern |
| 8 | U [1,3] | ||
| U [2J, 3J] | U [2,6] | ||
| Random{3, 4} | U 10 * [4,32] | ||
| Random{2, 3} | U 100 * [10,30] | ||
| U [0, 10] |
Table 3.
Type and dimension of test problems
| Size1 | Size2 | Size3 | Size4 | Size5 | Size6 | |
| (15*40*28) | (18*49*33) | (21*55*39) | (24*60*46) | (27*69*54) | (30*76*62) | |
| A1 | A2 | A3 | A4 | A5 | A6 | |
| B1 | B2 | B3 | B4 | B5 | B6 |
| Size1 | Size2 | Size3 | Size4 | Size5 | Size6 | |
| (15*40*28) | (18*49*33) | (21*55*39) | (24*60*46) | (27*69*54) | (30*76*62) | |
| A1 | A2 | A3 | A4 | A5 | A6 | |
| B1 | B2 | B3 | B4 | B5 | B6 |
Table 4.
The Pareto-optimal solutions for problems A3
| Solution number | Lévy-NSGA-Ⅱ | Solution number | MOCS | Solution number | NSGA-Ⅱ | |||
| Dissimilarity coefficient | Workload balance | Dissimilarity coefficient | Workload balance | Dissimilarity coefficient | Workload balance | |||
| 1 | 7.533333 | 2.022256 | 1 | 7.939286 | 1.976695 | 1 | 7.804762 | 2.074396 |
| 2 | 7.719048 | 1.921483 | 2 | 8.025000 | 1.830073 | 2 | 7.829762 | 1.976695 |
| 3 | 7.833333 | 1.855261 | 3 | 8.092063 | 1.802253 | 3 | 7.876190 | 1.918146 |
| 4 | 7.901190 | 1.851695 | 4 | 8.177778 | 1.629786 | 4 | 7.954762 | 1.830073 |
| 5 | 7.933730 | 1.835300 | 5 | 8.344444 | 1.503174 | 5 | 8.005159 | 1.694708 |
| 6 | 7.954762 | 1.830073 | 6 | 8.563492 | 1.457544 | 6 | 8.229762 | 1.661732 |
| 7 | 8.005159 | 1.694708 | 7 | 8.683730 | 1.391009 | 7 | 8.308333 | 1.653314 |
| 8 | 8.254762 | 1.629786 | 8 | 8.790476 | 1.266009 | 8 | 8.379762 | 1.559564 |
| 9 | 8.282143 | 1.490863 | 9 | 9.265476 | 1.224086 | 9 | 8.430159 | 1.503174 |
| 10 | 8.560714 | 1.457544 | 10 | 9.383333 | 1.167368 | 10 | 8.626190 | 1.470198 |
| 11 | 8.711111 | 1.391009 | 11 | 9.487302 | 1.131726 | 11 | 8.656349 | 1.391009 |
| 12 | 8.717063 | 1.297040 | 12 | 8.727778 | 1.266009 | |||
| 13 | 8.727778 | 1.266009 | 13 | 9.080556 | 1.224086 | |||
| 14 | 9.059524 | 1.224086 | 14 | 9.364286 | 1.131726 | |||
| 15 | 9.255952 | 1.167368 | ||||||
| 16 | 9.267857 | 1.131726 | ||||||
| Solution number | Lévy-NSGA-Ⅱ | Solution number | MOCS | Solution number | NSGA-Ⅱ | |||
| Dissimilarity coefficient | Workload balance | Dissimilarity coefficient | Workload balance | Dissimilarity coefficient | Workload balance | |||
| 1 | 7.533333 | 2.022256 | 1 | 7.939286 | 1.976695 | 1 | 7.804762 | 2.074396 |
| 2 | 7.719048 | 1.921483 | 2 | 8.025000 | 1.830073 | 2 | 7.829762 | 1.976695 |
| 3 | 7.833333 | 1.855261 | 3 | 8.092063 | 1.802253 | 3 | 7.876190 | 1.918146 |
| 4 | 7.901190 | 1.851695 | 4 | 8.177778 | 1.629786 | 4 | 7.954762 | 1.830073 |
| 5 | 7.933730 | 1.835300 | 5 | 8.344444 | 1.503174 | 5 | 8.005159 | 1.694708 |
| 6 | 7.954762 | 1.830073 | 6 | 8.563492 | 1.457544 | 6 | 8.229762 | 1.661732 |
| 7 | 8.005159 | 1.694708 | 7 | 8.683730 | 1.391009 | 7 | 8.308333 | 1.653314 |
| 8 | 8.254762 | 1.629786 | 8 | 8.790476 | 1.266009 | 8 | 8.379762 | 1.559564 |
| 9 | 8.282143 | 1.490863 | 9 | 9.265476 | 1.224086 | 9 | 8.430159 | 1.503174 |
| 10 | 8.560714 | 1.457544 | 10 | 9.383333 | 1.167368 | 10 | 8.626190 | 1.470198 |
| 11 | 8.711111 | 1.391009 | 11 | 9.487302 | 1.131726 | 11 | 8.656349 | 1.391009 |
| 12 | 8.717063 | 1.297040 | 12 | 8.727778 | 1.266009 | |||
| 13 | 8.727778 | 1.266009 | 13 | 9.080556 | 1.224086 | |||
| 14 | 9.059524 | 1.224086 | 14 | 9.364286 | 1.131726 | |||
| 15 | 9.255952 | 1.167368 | ||||||
| 16 | 9.267857 | 1.131726 | ||||||
Table 5.
Comparisons of 6 sets of test problems
| Problem No | Pareto distance |
Pareto distance |
Pareto distance |
||||||
| MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | |
| A1 | 0.1016 | 0.0484 | 0.0158 | 6 | 8 | 10 | 0.0367 | 0.0333 | 0.0333 |
| A2 | 0.0754 | 0.0346 | 0.0286 | 2 | 6 | 10 | 0.0367 | 0.0400 | 0.0333 |
| A3 | 0.1460 | 0.0644 | 0.0030 | 1 | 4 | 14 | 0.0367 | 0.0467 | 0.0533 |
| A4 | 0.2443 | 0.5967 | 0.0014 | 0 | 1 | 10 | 0.0367 | 0.0267 | 0.0400 |
| A5 | 2.0610 | 2.3134 | 0.0001 | 0 | 2 | 18 | 0.0367 | 0.0233 | 0.0633 |
| A6 | 2.3025 | 2.7586 | 0.0000 | 0 | 0 | 16 | 0.0500 | 0.0500 | 0.0533 |
| B1 | 0.4266 | 1.5974 | 0.0506 | 32 | 27 | 42 | 0.1333 | 0.1000 | 0.1467 |
| B2 | 1.0217 | 2.0321 | 0.0123 | 18 | 18 | 51 | 0.1067 | 0.1167 | 0.1800 |
| B3 | 0.3373 | 0.4506 | 0.2616 | 50 | 67 | 107 | 0.2600 | 0.3000 | 0.3900 |
| B4 | 0.7522 | 1.6053 | 0.5043 | 48 | 20 | 67 | 0.3600 | 0.3633 | 0.2800 |
| B5 | 1.5456 | 3.0284 | 0.4457 | 26 | 43 | 42 | 0.2267 | 0.2233 | 0.2600 |
| B6 | 0.8931 | 2.0602 | 0.3225 | 29 | 16 | 100 | 0.2700 | 0.2400 | 0.4267 |
| Problem No | Pareto distance |
Pareto distance |
Pareto distance |
||||||
| MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | MOCS | NSGA-Ⅱ | Lévy-NSGA-Ⅱ | |
| A1 | 0.1016 | 0.0484 | 0.0158 | 6 | 8 | 10 | 0.0367 | 0.0333 | 0.0333 |
| A2 | 0.0754 | 0.0346 | 0.0286 | 2 | 6 | 10 | 0.0367 | 0.0400 | 0.0333 |
| A3 | 0.1460 | 0.0644 | 0.0030 | 1 | 4 | 14 | 0.0367 | 0.0467 | 0.0533 |
| A4 | 0.2443 | 0.5967 | 0.0014 | 0 | 1 | 10 | 0.0367 | 0.0267 | 0.0400 |
| A5 | 2.0610 | 2.3134 | 0.0001 | 0 | 2 | 18 | 0.0367 | 0.0233 | 0.0633 |
| A6 | 2.3025 | 2.7586 | 0.0000 | 0 | 0 | 16 | 0.0500 | 0.0500 | 0.0533 |
| B1 | 0.4266 | 1.5974 | 0.0506 | 32 | 27 | 42 | 0.1333 | 0.1000 | 0.1467 |
| B2 | 1.0217 | 2.0321 | 0.0123 | 18 | 18 | 51 | 0.1067 | 0.1167 | 0.1800 |
| B3 | 0.3373 | 0.4506 | 0.2616 | 50 | 67 | 107 | 0.2600 | 0.3000 | 0.3900 |
| B4 | 0.7522 | 1.6053 | 0.5043 | 48 | 20 | 67 | 0.3600 | 0.3633 | 0.2800 |
| B5 | 1.5456 | 3.0284 | 0.4457 | 26 | 43 | 42 | 0.2267 | 0.2233 | 0.2600 |
| B6 | 0.8931 | 2.0602 | 0.3225 | 29 | 16 | 100 | 0.2700 | 0.2400 | 0.4267 |
Table 6.
Parts information for the numerical example
| Parts | Routes | Operation | Demand | Time | Parts | Routes | Operation | Demand | Time |
| P1 | R1 | 1 | 5, 7 | 360 | P7 | R3 | 2 | 90 | |
| 8 | 90 | 8 | 120 | ||||||
| 6 | 160 | 5 | 160 | ||||||
| 2 | 180 | 2 | 50 | ||||||
| R2 | 8 | 90 | P8 | R1 | 1 | 4, 7 | 360 | ||
| 5 | 180 | 7 | 90 | ||||||
| 1 | 360 | 2 | 160 | ||||||
| 6 | 160 | 4 | 180 | ||||||
| 2 | 180 | 6 | 90 | ||||||
| P2 | R1 | 2 | 10, 8 | 180 | R2 | 2 | 180 | ||
| 6 | 80 | 8 | 360 | ||||||
| 4 | 120 | 5 | 160 | ||||||
| 7 | 90 | 2 | 340 | ||||||
| R2 | 1 | 160 | P9 | R1 | 1 | 7, 5 | 360 | ||
| 5 | 90 | 5 | 160 | ||||||
| 3 | 120 | 2 | 180 | ||||||
| 7 | 90 | 3 | 190 | ||||||
| R3 | 4 | 120 | 7 | 80 | |||||
| 1 | 160 | 1 | 160 | ||||||
| 5 | 60 | R2 | 1 | 360 | |||||
| 3 | 90 | 8 | 100 | ||||||
| P3 | R1 | 8 | 0, 5 | 80 | 2 | 160 | |||
| 1 | 160 | 4 | 180 | ||||||
| 6 | 80 | 6 | 90 | ||||||
| 2 | 200 | 2 | 180 | ||||||
| P4 | R1 | 2 | 5, 0 | 120 | P10 | R1 | 4 | 0, 10 | 90 |
| 6 | 100 | 8 | 100 | ||||||
| 1 | 200 | 2 | 120 | ||||||
| 4 | 140 | 5 | 60 | ||||||
| 1 | 150 | R2 | 6 | 60 | |||||
| P5 | R1 | 2 | 6, 10 | 120 | 1 | 140 | |||
| 3 | 60 | 7 | 100 | ||||||
| 7 | 140 | 3 | 80 | ||||||
| R2 | 1 | 120 | R3 | 5 | 60 | ||||
| 4 | 60 | 2 | 100 | ||||||
| 7 | 120 | 7 | 100 | ||||||
| 5 | 120 | 3 | 70 | ||||||
| P6 | R1 | 2 | 4, 0 | 120 | P11 | R1 | 2 | 10, 0 | 360 |
| 7 | 60 | 3 | 90 | ||||||
| 6 | 100 | 7 | 120 | ||||||
| 2 | 120 | 1 | 180 | ||||||
| P7 | R1 | 1 | 6, 4 | 90 | 4 | 90 | |||
| 5 | 160 | 7 | 40 | ||||||
| 2 | 50 | 5 | 360 | ||||||
| 7 | 120 | P12 | R1 | 4 | 9, 8 | 360 | |||
| R2 | 2 | 120 | 6 | 90 | |||||
| 8 | 60 | 2 | 300 | ||||||
| 6 | 80 | 8 | 180 | ||||||
| 2 | 120 | 5 | 90 |
| Parts | Routes | Operation | Demand | Time | Parts | Routes | Operation | Demand | Time |
| P1 | R1 | 1 | 5, 7 | 360 | P7 | R3 | 2 | 90 | |
| 8 | 90 | 8 | 120 | ||||||
| 6 | 160 | 5 | 160 | ||||||
| 2 | 180 | 2 | 50 | ||||||
| R2 | 8 | 90 | P8 | R1 | 1 | 4, 7 | 360 | ||
| 5 | 180 | 7 | 90 | ||||||
| 1 | 360 | 2 | 160 | ||||||
| 6 | 160 | 4 | 180 | ||||||
| 2 | 180 | 6 | 90 | ||||||
| P2 | R1 | 2 | 10, 8 | 180 | R2 | 2 | 180 | ||
| 6 | 80 | 8 | 360 | ||||||
| 4 | 120 | 5 | 160 | ||||||
| 7 | 90 | 2 | 340 | ||||||
| R2 | 1 | 160 | P9 | R1 | 1 | 7, 5 | 360 | ||
| 5 | 90 | 5 | 160 | ||||||
| 3 | 120 | 2 | 180 | ||||||
| 7 | 90 | 3 | 190 | ||||||
| R3 | 4 | 120 | 7 | 80 | |||||
| 1 | 160 | 1 | 160 | ||||||
| 5 | 60 | R2 | 1 | 360 | |||||
| 3 | 90 | 8 | 100 | ||||||
| P3 | R1 | 8 | 0, 5 | 80 | 2 | 160 | |||
| 1 | 160 | 4 | 180 | ||||||
| 6 | 80 | 6 | 90 | ||||||
| 2 | 200 | 2 | 180 | ||||||
| P4 | R1 | 2 | 5, 0 | 120 | P10 | R1 | 4 | 0, 10 | 90 |
| 6 | 100 | 8 | 100 | ||||||
| 1 | 200 | 2 | 120 | ||||||
| 4 | 140 | 5 | 60 | ||||||
| 1 | 150 | R2 | 6 | 60 | |||||
| P5 | R1 | 2 | 6, 10 | 120 | 1 | 140 | |||
| 3 | 60 | 7 | 100 | ||||||
| 7 | 140 | 3 | 80 | ||||||
| R2 | 1 | 120 | R3 | 5 | 60 | ||||
| 4 | 60 | 2 | 100 | ||||||
| 7 | 120 | 7 | 100 | ||||||
| 5 | 120 | 3 | 70 | ||||||
| P6 | R1 | 2 | 4, 0 | 120 | P11 | R1 | 2 | 10, 0 | 360 |
| 7 | 60 | 3 | 90 | ||||||
| 6 | 100 | 7 | 120 | ||||||
| 2 | 120 | 1 | 180 | ||||||
| P7 | R1 | 1 | 6, 4 | 90 | 4 | 90 | |||
| 5 | 160 | 7 | 40 | ||||||
| 2 | 50 | 5 | 360 | ||||||
| 7 | 120 | P12 | R1 | 4 | 9, 8 | 360 | |||
| R2 | 2 | 120 | 6 | 90 | |||||
| 8 | 60 | 2 | 300 | ||||||
| 6 | 80 | 8 | 180 | ||||||
| 2 | 120 | 5 | 90 |
Table 7.
Process-machine incidence matrix for the numerical example
| Operation | Machine type | Machine name | Machine number | Available processing time /min |
| 1 | M1 | CNC lathes | 5 | 3000 |
| 2 | M2 | Ordinary lathes | 5 | 3000 |
| 3 | M3 | Slotting machines | 2 | 1700 |
| 4 | M4 | CNC slotting machines | 5 | 1600 |
| 5 | M5 | Grinders | 6 | 1200 |
| 6 | M6 | Grinding machines | 4 | 1200 |
| 7 | M7 | Gun Drill | 3 | 1600 |
| 8 | M8 | Drilling machines | 3 | 1200 |
| Operation | Machine type | Machine name | Machine number | Available processing time /min |
| 1 | M1 | CNC lathes | 5 | 3000 |
| 2 | M2 | Ordinary lathes | 5 | 3000 |
| 3 | M3 | Slotting machines | 2 | 1700 |
| 4 | M4 | CNC slotting machines | 5 | 1600 |
| 5 | M5 | Grinders | 6 | 1200 |
| 6 | M6 | Grinding machines | 4 | 1200 |
| 7 | M7 | Gun Drill | 3 | 1600 |
| 8 | M8 | Drilling machines | 3 | 1200 |
Table 8.
One of the schemes for the first period of the numerical example
| Cell | Part(routing) | Number of each machine type in each cell | f1 | f2 | |||||||
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | ||||
| 1 | 1(2), 4(1), 8(1), 9(2), 12(1) | 3 | 3 | 0 | 4 | 2 | 3 | 1 | 3 | 4.0190 | 0.5939 |
| 2 | 2(2), 5(2), 11(1) | 2 | 2 | 2 | 1 | 5 | 0 | 3 | 0 | ||
| 3 | 6(1), 7(2) | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | ||
| Cell | Part(routing) | Number of each machine type in each cell | f1 | f2 | |||||||
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | ||||
| 1 | 1(2), 4(1), 8(1), 9(2), 12(1) | 3 | 3 | 0 | 4 | 2 | 3 | 1 | 3 | 4.0190 | 0.5939 |
| 2 | 2(2), 5(2), 11(1) | 2 | 2 | 2 | 1 | 5 | 0 | 3 | 0 | ||
| 3 | 6(1), 7(2) | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | ||
Table 9.
One of the schemes for the second period of the numerical example
| Number of each machine type in each cell | f1 | f2 | f3 | |||||||||
| Cell | Part(routing) | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |||
| 1 | 1(2), 8(1), 9(2), 12(1) | 3 | 3 | 0 | 4 | 2 | 3 | 1 | 3 | 3.3619 | 0.6384 | 6 |
| 2 | 2(2), 5(2), 7(1), 10(3) | 1 | 1 | 1 | 1 | 3 | 0 | 3 | 0 | |||
| 3 | 3(1) | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | |||
| Number of each machine type in each cell | f1 | f2 | f3 | |||||||||
| Cell | Part(routing) | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |||
| 1 | 1(2), 8(1), 9(2), 12(1) | 3 | 3 | 0 | 4 | 2 | 3 | 1 | 3 | 3.3619 | 0.6384 | 6 |
| 2 | 2(2), 5(2), 7(1), 10(3) | 1 | 1 | 1 | 1 | 3 | 0 | 3 | 0 | |||
| 3 | 3(1) | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | |||
Table 10.
Detailed machine changes between two formation periods
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
| Cell 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Cell 2 | -1 | -1 | -1 | 0 | -2 | 0 | 0 | 0 |
| Cell 3 | +1 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
| Cell 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Cell 2 | -1 | -1 | -1 | 0 | -2 | 0 | 0 | 0 |
| Cell 3 | +1 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
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