doi: 10.3934/jimo.2022009
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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

*Corresponding author: Lei Yue

Received  July 2021 Revised  October 2021 Early access January 2022

Fund Project: The authors are supported by the Guangdong Province Key Field R & D Program (2020B0101050001) and the National Natural Science Foundation of China (No. 51905196)

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.

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
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. BortoliniE. FerrariF. 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. ChouC. 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. DebA. PratapS. 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. EguiaJ. C. MolinaS. 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. ErfaniS. 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. FengW. DaL. XiE. 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. FungF. LiangZ. 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. GuoM. ChenK. MohamedT. QuS. 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. HachichaF. 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. HamediG. EsmaeilianN. 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. HanY. YuL. GaoJ. 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. HosseiniM. M. PaydarI. 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. ImranC. KangY. H. LeeM. 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. LiA. 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. LiQ. 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. LiQ. 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. LiH. Y. SangY. Y. HanC. 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. LiuJ. 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. MahdaviA. AalaeiM. 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. MoradgholiM. M. PaydarI. 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. NiakanA. BaboliT. 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. PanL. 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. RabbaniH. 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. RahimiJ. 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. RezaeianN. JavadianR. 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. ViswanathanS. V. BuldyrevS. HavlinM. Da LuzE. 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. BortoliniE. FerrariF. 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. ChouC. 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. DebA. PratapS. 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. EguiaJ. C. MolinaS. 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. ErfaniS. 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. FengW. DaL. XiE. 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. FungF. LiangZ. 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. GuoM. ChenK. MohamedT. QuS. 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. HachichaF. 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. HamediG. EsmaeilianN. 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. HanY. YuL. GaoJ. 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. HosseiniM. M. PaydarI. 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. ImranC. KangY. H. LeeM. 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. LiA. 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. LiQ. 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. LiQ. 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. LiH. Y. SangY. Y. HanC. 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. LiuJ. 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. MahdaviA. AalaeiM. 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. MoradgholiM. M. PaydarI. 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. NiakanA. BaboliT. 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. PanL. 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. RabbaniH. 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. RahimiJ. 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. RezaeianN. JavadianR. 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. ViswanathanS. V. BuldyrevS. HavlinM. Da LuzE. 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 1.  The dynamic virtual cell reconfiguration framework under multiple planning cycles
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
$ j $ part types, $ j=1, 2, \cdots, J $
$ r $ process routings, $ r =1, 2, \cdots, R_j $
$ m $ machine types, $ m=1, 2, \cdots, M $
$ c $ virtual cells, $ c=1, 2, \cdots, C $
$ t $ formation periods
Input parameters
$ J_t $ number of part types in period $ t $
$ R_j $ number of process routings for part type $ j $
M number of machine types
$ N_m $ number of machines included in machine type $ m $
$ B_U $ upper bounds of virtual cells
$ B_L $ lower bounds of virtual cells
$ D_{j, t} $ demand for part type $ j $ in period $ t $
$ A_m $ production capacity of each machine of type $ m $
$ \alpha_{j, r, m} $ 1, if $ r $-th process routing of part type $ j $ needs to use the machine type $ m $, 0 otherwise
$ T_{j, r, m} $ processing time of $ r $-th process routing of part type $ j $ at machine type $ m $
$ S_{j, j', r, r'} $ the similarity coefficient between $ r $-th process routing of part type $ j $ and $ r' $-th process routing of part type $ j' $
$ Z_{m, c, t-1} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t-1\ (t>1) $
Variables
$ C_t $ number of virtual cells in period $ t $, $ B_L\leq C \leq B_U $
$ X_{j, r, c, t} $ 1, part type $ j $ to be assigned to routing $ r $ and to be assigned to virtual cell $ c $ in period $ t $, 0 otherwise
$ Y_{m, c, t} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t $ (real number)
$ Z_{m, c, t} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t $ (integer)
Indices
$ j $ part types, $ j=1, 2, \cdots, J $
$ r $ process routings, $ r =1, 2, \cdots, R_j $
$ m $ machine types, $ m=1, 2, \cdots, M $
$ c $ virtual cells, $ c=1, 2, \cdots, C $
$ t $ formation periods
Input parameters
$ J_t $ number of part types in period $ t $
$ R_j $ number of process routings for part type $ j $
M number of machine types
$ N_m $ number of machines included in machine type $ m $
$ B_U $ upper bounds of virtual cells
$ B_L $ lower bounds of virtual cells
$ D_{j, t} $ demand for part type $ j $ in period $ t $
$ A_m $ production capacity of each machine of type $ m $
$ \alpha_{j, r, m} $ 1, if $ r $-th process routing of part type $ j $ needs to use the machine type $ m $, 0 otherwise
$ T_{j, r, m} $ processing time of $ r $-th process routing of part type $ j $ at machine type $ m $
$ S_{j, j', r, r'} $ the similarity coefficient between $ r $-th process routing of part type $ j $ and $ r' $-th process routing of part type $ j' $
$ Z_{m, c, t-1} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t-1\ (t>1) $
Variables
$ C_t $ number of virtual cells in period $ t $, $ B_L\leq C \leq B_U $
$ X_{j, r, c, t} $ 1, part type $ j $ to be assigned to routing $ r $ and to be assigned to virtual cell $ c $ in period $ t $, 0 otherwise
$ Y_{m, c, t} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t $ (real number)
$ Z_{m, c, t} $ number of machines of type $ m $ assigned to virtual cell $ c $ in period $ t $ (integer)
Table 2.  Pattern of data generation
Parameter Generation pattern Parameter Generation pattern
$ M $ 8 $ R_j $ U [1,3]
$ \sum_{m=1}^{M}{N_m} $ U [2J, 3J] $ \sum_{m=1}^{M}{\alpha_{j, r, m}} $ U [2,6]
$ B_U $ Random{3, 4} $ T_{j, r, m} $ U 10 * [4,32]
$ B_L $ Random{2, 3} $ A_m $ U 100 * [10,30]
$ D_j $ U [0, 10]
Parameter Generation pattern Parameter Generation pattern
$ M $ 8 $ R_j $ U [1,3]
$ \sum_{m=1}^{M}{N_m} $ U [2J, 3J] $ \sum_{m=1}^{M}{\alpha_{j, r, m}} $ U [2,6]
$ B_U $ Random{3, 4} $ T_{j, r, m} $ U 10 * [4,32]
$ B_L $ Random{2, 3} $ A_m $ U 100 * [10,30]
$ D_j $ 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)
$ T=1 $ A1 A2 A3 A4 A5 A6
$ T>1 $ 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)
$ T=1 $ A1 A2 A3 A4 A5 A6
$ T>1 $ 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 $ V_{pd} $ Pareto distance $ V_{np} $ Pareto distance $ V_{rd} $
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 $ V_{pd} $ Pareto distance $ V_{np} $ Pareto distance $ V_{rd} $
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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