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doi: 10.3934/jimo.2018007

A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment

1. 

Department of Industrial Engineering, Yalova University, Yalova, Turkey

2. 

Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

* Corresponding author: Omer Faruk Yilmaz

Received  August 2016 Revised  June 2017 Published  January 2018

This paper focuses on the batch scheduling problem in multi-hybrid cell manufacturing systems (MHCMS) in a dual-resource constrained (DRC) setting, considering skilled workforce assignment (SWA). This problem consists of finding the sequence of batches on each cell, the starting time of each batch, and assigning employees to the operations of batches in accordance with the desired objective. Because handling both the scheduling and assignment decisions simultaneously presents a challenging optimization problem, it is difficult to solve the formulated model, even for small-sized problem instances. Three metaheuristics are proposed to solve the batch scheduling problem, namely the genetic algorithm (GA), simulated annealing (SA) algorithm, and artificial bee colony (ABC) algorithm. A serial scheduling scheme (SSS) is introduced and employed in all metaheuristics to obtain a feasible schedule for each individual. The main aim of this study is to identify an effective metaheuristic for determining the scheduling and assignment decisions that minimize the average cell response time. Detailed computational experiments were conducted, based on real production data, to evaluate the performance of the metaheuristics. The experimental results show that the performance of the proposed ABC algorithm is superior to other metaheuristics for different levels of experimental factors determined for the number of batches and the employee flexibility.

Citation: Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018007
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A. AzadehM. Sheikhalishahi and M. Koushan, An integrated fuzzy DEA-Fuzzy simulation approach for optimization of operator allocation with learning effects in multi products CMS, Applied Mathematical Modelling, 37 (2013), 9922-9933. doi: 10.1016/j.apm.2013.05.039.

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A. N. Balaji and S. Porselvi, Artificial immune system algorithm and simulated annealing algorithm for scheduling batches of parts based on job availability model in a multi-cell flexible manufacturing system, Procedia Engineering, 97 (2014), 1524-1533. doi: 10.1016/j.proeng.2014.12.436.

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V. I. Cesani and H. J. Steudel, A study of labor assignment flexibility in cellular manufacturing systems, Computers & Industrial Engineering, 48 (2005), 571-591. doi: 10.1016/j.cie.2003.04.001.

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W. D. Chang, Nonlinear CSTR control system design using an artificial bee colony algorithm, Simulation Modelling Practice and Theory, 31 (2013), 1-9. doi: 10.1016/j.simpat.2012.11.002.

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W. D. Chang, Equation of state calculations by fast computing machines, The Journal of Chemical Physics, 21 (1953), 1087-1092.

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E. B. EdisC. Oguz and I. Ozkarahan, Parallel machine scheduling with additional resources: Notation, classification, models and solution methods, European Journal of Operational Research, 230 (2013), 449-463. doi: 10.1016/j.ejor.2013.02.042.

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G. EgilmezB. Erenay and G. A. Suer, Stochastic skill-based manpower allocation in a cellular manufacturing system, Journal of Manufacturing Systems, 33 (2014), 578-588. doi: 10.1016/j.jmsy.2014.05.005.

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B. FahimniaH. Davarzani and A. Eshragh, Planning of complex supply chains: A performance comparison of three meta-heuristic algorithms, Computers & Operations Research, 89 (2018), 241-252. doi: 10.1016/j.cor.2015.10.008.

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J. FanM. Cao and D. Feng, Multi-objective dual resource-constrained model for cell formation problem, In Management of Innovation and Technology IEEE International Conference, (2010), 1031-1036. doi: 10.1109/ICMIT.2010.5492881.

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N. Karaboga, A new design method based on artificial bee colony algorithm for digital IIR filters, Journal of the Franklin Institute, 346 (2009), 328-348. doi: 10.1016/j.jfranklin.2008.11.003.

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D. Karaboga and B. Basturk, On the performance of artificial bee colony (ABC) algorithm, Applied Soft Computing, 8 (2008), 687-697. doi: 10.1016/j.asoc.2007.05.007.

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D. Karaboga and E. Kaya, An adaptive and hybrid artificial bee colony algorithm (aABC) for ANFIS training, Applied Soft Computing, 49 (2016), 423-436. doi: 10.1016/j.asoc.2016.07.039.

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[35]

X. Li and L. Gao, An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem, International Journal of Production Economics, 174 (2016), 93-110. doi: 10.1016/j.ijpe.2016.01.016.

[36]

Y. LiX. Li and J. N. D. Gupta, Solving the multi-objective flowline manufacturing cell scheduling problem by hybrid harmony search, Expert Systems with Applications, 42 (2015), 1409-1417. doi: 10.1016/j.eswa.2014.09.007.

[37]

C. LiuJ. Wang and J. Y. T. Leung, Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm, Computers & Industrial Engineering, 96 (2016), 162-179. doi: 10.1016/j.cie.2016.03.020.

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D. NasoM. SuricoB. Turchiano and U. Kaymak, Genetic algorithms for supply-chain scheduling: A case study in the distribution of ready-mixed concrete, European Journal of Operational Research, 177 (2007), 2069-2099. doi: 10.1016/j.ejor.2005.12.019.

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F. NiakanA. BaboliT. Moyaux and V. Botta-Genoulaz, A new multi-objective mathematical model for dynamic cell formation under demand and cost uncertainty considering social criteria, Applied Mathematical Modelling, 40 (2016), 2674-2691. doi: 10.1016/j.apm.2015.09.047.

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I. Ribas and R. Companys, Efficient heuristic algorithms for the blocking flow shop scheduling problem with total flow time minimization, Computers & Industrial Engineering, 87 (2015), 30-39. doi: 10.1016/j.cie.2015.04.013.

[48]

S. I. SatogluM. B. Durmusoglu and T. Ertay, A mathematical model and a heuristic approach for design of the hybrid manufacturing systems to facilitate one-piece flow, International Journal of Production Research, 48 (2010), 5195-5220. doi: 10.1080/00207540903089544.

[49]

S. I. Satoglu and N. C. Suresh, A goal-programming approach for design of hybrid cellular manufacturing systems in dual resource constrained environments, Computers & Industrial Engineering, 56 (2009), 560-575. doi: 10.1016/j.cie.2008.06.009.

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J. E. SchallerJ. N. D Gupta and A. J. Vakharia, Scheduling a flowline manufacturing cell with sequence dependent family setup times, European Journal of Operational Research, 125 (2000), 324-339.

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T. SousaT. SoaresH. MoraisR. Castro and Z. Vale, Simulated annealing to handle energy and ancillary services joint management considering electric vehicles, Electric Power Systems Research, 136 (2016), 383-397. doi: 10.1016/j.epsr.2016.03.031.

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G. A. Suer and O. Alhawari, Operator assignment decisions in a highly dynamic cellular environment, Operations Management Research and Cellular Manufacturing Systems: Innovative Methods and Approaches, (2011), 258-294.

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G. A. Suer and C. Dagli, Intra-cell manpower transfers and cell loading in labor-intensive manufacturing cells, Computers & Industrial Engineering, 48 (2005), 643-655. doi: 10.1016/j.cie.2003.03.006.

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show all references

References:
[1]

M. O. Adamu and A. Adewumi, Comparing the performance of different meta-heuristics for unweighted parallel machine scheduling, South African Journal of Industrial Engineering, 26 (2015), 143-157. doi: 10.7166/26-2-628.

[2]

A. O. Adewumi and M. M. Ali, A multi-level genetic algorithm for a multi-stage space allocation problem, Mathematical and Computer Modeling, 51 (2010), 109-126. doi: 10.1016/j.mcm.2009.09.004.

[3]

A. O. SawyerrB. A. Adewumi and M. M. Ali, A heuristic solution to the university timetabling problem, Engineering Computations, 26 (1984), 972-984. doi: 10.1108/02644400910996853.

[4]

R. G. Askin and Y. Huang, Forming effective worker teams for cellular manufacturing, International Journal of Production Research, 39 (2001), 2431-2451. doi: 10.1080/00207540110040466.

[5]

A. AzadehM. Sheikhalishahi and M. Koushan, An integrated fuzzy DEA-Fuzzy simulation approach for optimization of operator allocation with learning effects in multi products CMS, Applied Mathematical Modelling, 37 (2013), 9922-9933. doi: 10.1016/j.apm.2013.05.039.

[6]

A. N. Balaji and S. Porselvi, Artificial immune system algorithm and simulated annealing algorithm for scheduling batches of parts based on job availability model in a multi-cell flexible manufacturing system, Procedia Engineering, 97 (2014), 1524-1533. doi: 10.1016/j.proeng.2014.12.436.

[7]

J. T. Black and S. L. Hunter, Lean Manufacturing Systems and Cell Design, Society of Manufacturing Engineers, 2003.

[8]

G. CelanoA. Costa and S. Fichera, Scheduling of unrelated parallel manufacturing cells with limited human resources, International Journal of Production Research, 46 (2008), 405-427. doi: 10.1080/00207540601138452.

[9]

V. I. Cesani and H. J. Steudel, A study of labor assignment flexibility in cellular manufacturing systems, Computers & Industrial Engineering, 48 (2005), 571-591. doi: 10.1016/j.cie.2003.04.001.

[10]

W. D. Chang, Nonlinear CSTR control system design using an artificial bee colony algorithm, Simulation Modelling Practice and Theory, 31 (2013), 1-9. doi: 10.1016/j.simpat.2012.11.002.

[11]

W. D. Chang, Equation of state calculations by fast computing machines, The Journal of Chemical Physics, 21 (1953), 1087-1092.

[12]

S. Chetty and A. O. Adewumi, Three new stochastic local search metaheuristics for the annual crop planning problem based on a new irrigation scheme, Journal of Applied Mathematics, 2013 (2013), Article ID 158538, 14 pages. doi: 10.1155/2013/158538.

[13]

D. A. Coley, An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific Pub. Co. Inc., 1999. doi: 10.1142/3904.

[14]

P. CortesJ. LarranetaL. OnievaJ. M. García and M. S. Caraballo, Genetic algorithm for planning cable telecommunication networks, Applied Soft Computing, 1 (2001), 21-33. doi: 10.1016/S1568-4946(01)00004-7.

[15]

A. CostaF. A. Cappadonna and S. Fichera, Joint optimization of a flow-shop group scheduling with sequence dependent set-up times and skilled workforce assignment, International Journal of Production Research, 52 (2014), 2696-2728. doi: 10.1080/00207543.2014.883469.

[16]

A. CostaF. A. Cappadonna and S. Fichera, A hybrid genetic algorithm for job sequencing and worker allocation in parallel unrelated machines with sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 69 (2013), 2799-2817. doi: 10.1007/s00170-013-5221-5.

[17]

A. CostaS. Fichera and F. A. Cappadonna, A genetic algorithm for scheduling both jobs families and skilled workforce, International Journal of Operations and Quantitative Management, 19 (2013), 221-247.

[18]

R. L. DanielsB. J. Hoopes and J. B. Mazzola, Scheduling parallel manufacturing cells with resource flexibility, Management Science, 42 (1996), 1260-1276. doi: 10.1287/mnsc.42.9.1260.

[19]

S. R. Das and C. Canel, An algorithm for scheduling batches of parts in a multi-cell flexible manufacturing system, International Journal of Production Economics, 97 (2005), 247-262. doi: 10.1016/j.ijpe.2004.07.006.

[20]

D. J. DavisH. V. Kher and B. J. Wagner, Influence of workload imbalances on the need for worker flexibility, Computers & Industrial Engineering, 57 (2009), 319-329. doi: 10.1016/j.cie.2008.11.029.

[21]

E. B. EdisC. Oguz and I. Ozkarahan, Parallel machine scheduling with additional resources: Notation, classification, models and solution methods, European Journal of Operational Research, 230 (2013), 449-463. doi: 10.1016/j.ejor.2013.02.042.

[22]

G. EgilmezB. Erenay and G. A. Suer, Stochastic skill-based manpower allocation in a cellular manufacturing system, Journal of Manufacturing Systems, 33 (2014), 578-588. doi: 10.1016/j.jmsy.2014.05.005.

[23]

B. FahimniaH. Davarzani and A. Eshragh, Planning of complex supply chains: A performance comparison of three meta-heuristic algorithms, Computers & Operations Research, 89 (2018), 241-252. doi: 10.1016/j.cor.2015.10.008.

[24]

J. FanM. Cao and D. Feng, Multi-objective dual resource-constrained model for cell formation problem, In Management of Innovation and Technology IEEE International Conference, (2010), 1031-1036. doi: 10.1109/ICMIT.2010.5492881.

[25]

J. W. FowlerP. Wirojanagud and E. S. Gel, Heuristics for workforce planning with worker differences, European Journal of Operational Research, 190 (2008), 724-740. doi: 10.1016/j.ejor.2007.06.038.

[26]

J. H. Holland, Genetic algorithms, Scientific American, 267 (1992), 66-72.

[27]

M. P. Hottenstein and S. A. Bowman, Cross-training and worker flexibility: A review of DRC system research, The Journal of High Technology Management Research, 9 (1998), 157-174. doi: 10.1016/S1047-8310(98)90002-5.

[28]

H. Hyer and U. Wemmerlov, Reorganizing the Factory Competing Through Cellular Manufacturing, Productivity Press, 2002.

[29]

J. B. Jensen, The impact of resource flexibility and staffing decisions on cellular and departmental shop performance, European Journal of Operational Research, 127 (2000), 279-296. doi: 10.1016/S0377-2217(99)00491-9.

[30]

N. Karaboga, A new design method based on artificial bee colony algorithm for digital IIR filters, Journal of the Franklin Institute, 346 (2009), 328-348. doi: 10.1016/j.jfranklin.2008.11.003.

[31]

D. Karaboga and B. Basturk, On the performance of artificial bee colony (ABC) algorithm, Applied Soft Computing, 8 (2008), 687-697. doi: 10.1016/j.asoc.2007.05.007.

[32]

D. Karaboga and E. Kaya, An adaptive and hybrid artificial bee colony algorithm (aABC) for ANFIS training, Applied Soft Computing, 49 (2016), 423-436. doi: 10.1016/j.asoc.2016.07.039.

[33]

S. Kirkpatrick, Optimization by simulated annealing: Quantitative studies, Journal of Statistical Physics, 34 (1984), 975-986. doi: 10.1007/BF01009452.

[34]

R. Kolisch, Serial and parallel resource-constrained project scheduling methods revisited: Theory and computation, European Journal of Operational Research, 90 (1996), 320-333. doi: 10.1016/0377-2217(95)00357-6.

[35]

X. Li and L. Gao, An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem, International Journal of Production Economics, 174 (2016), 93-110. doi: 10.1016/j.ijpe.2016.01.016.

[36]

Y. LiX. Li and J. N. D. Gupta, Solving the multi-objective flowline manufacturing cell scheduling problem by hybrid harmony search, Expert Systems with Applications, 42 (2015), 1409-1417. doi: 10.1016/j.eswa.2014.09.007.

[37]

C. LiuJ. Wang and J. Y. T. Leung, Worker assignment and production planning with learning and forgetting in manufacturing cells by hybrid bacteria foraging algorithm, Computers & Industrial Engineering, 96 (2016), 162-179. doi: 10.1016/j.cie.2016.03.020.

[38]

K. F. Man, K. S. Tang and S. Kwong, Genetic Algorithms: Concepts and Designs, Springer-Verlag London, Ltd., London, 1999.

[39]

B. L. Miller and D. E. Goldberg, Genetic algorithms, tournament selection, and the effects of noise, Complex Systems, 9 (1995), 193-212.

[40]

J. Miltenburg, One-piece flow manufacturing on U-shaped production lines: A tutorial, IIE Transactions, 33 (2001), 303-321. doi: 10.1080/07408170108936831.

[41]

D. NasoM. SuricoB. Turchiano and U. Kaymak, Genetic algorithms for supply-chain scheduling: A case study in the distribution of ready-mixed concrete, European Journal of Operational Research, 177 (2007), 2069-2099. doi: 10.1016/j.ejor.2005.12.019.

[42]

F. NiakanA. BaboliT. Moyaux and V. Botta-Genoulaz, A new multi-objective mathematical model for dynamic cell formation under demand and cost uncertainty considering social criteria, Applied Mathematical Modelling, 40 (2016), 2674-2691. doi: 10.1016/j.apm.2015.09.047.

[43]

B. A. NormanW. TharmmaphornphilasK. L. NeedyB. Bidanda and R. C. Warner, Worker assignment in cellular manufacturing considering technical and human skills, International Journal of Production Research, 40 (2002), 1479-1492. doi: 10.1080/00207540110118082.

[44]

M. W. Park and Y. D. Kim, A systematic procedure for setting parameters in simulated annealing algorithms, Computers & Operations Research, 25 (1998), 207-217. doi: 10.1016/S0305-0548(97)00054-3.

[45]

D. Quagliarella, Genetic Algorithms and Evolution Strategy in Engineering and Computer Science: Recent Advances and Industrial Applications, World John Wiley & Son Ltd., 1998.

[46]

S. RameshS. Kannan and S. Baskar, Application of modified NSGA-II algorithm to multi-objective reactive power planning, Applied Soft Computing, 12 (2012), 741-753. doi: 10.1016/j.asoc.2011.09.015.

[47]

I. Ribas and R. Companys, Efficient heuristic algorithms for the blocking flow shop scheduling problem with total flow time minimization, Computers & Industrial Engineering, 87 (2015), 30-39. doi: 10.1016/j.cie.2015.04.013.

[48]

S. I. SatogluM. B. Durmusoglu and T. Ertay, A mathematical model and a heuristic approach for design of the hybrid manufacturing systems to facilitate one-piece flow, International Journal of Production Research, 48 (2010), 5195-5220. doi: 10.1080/00207540903089544.

[49]

S. I. Satoglu and N. C. Suresh, A goal-programming approach for design of hybrid cellular manufacturing systems in dual resource constrained environments, Computers & Industrial Engineering, 56 (2009), 560-575. doi: 10.1016/j.cie.2008.06.009.

[50]

J. E. SchallerJ. N. D Gupta and A. J. Vakharia, Scheduling a flowline manufacturing cell with sequence dependent family setup times, European Journal of Operational Research, 125 (2000), 324-339.

[51]

T. SousaT. SoaresH. MoraisR. Castro and Z. Vale, Simulated annealing to handle energy and ancillary services joint management considering electric vehicles, Electric Power Systems Research, 136 (2016), 383-397. doi: 10.1016/j.epsr.2016.03.031.

[52]

G. A. Suer and O. Alhawari, Operator assignment decisions in a highly dynamic cellular environment, Operations Management Research and Cellular Manufacturing Systems: Innovative Methods and Approaches, (2011), 258-294.

[53]

G. A. Suer and C. Dagli, Intra-cell manpower transfers and cell loading in labor-intensive manufacturing cells, Computers & Industrial Engineering, 48 (2005), 643-655. doi: 10.1016/j.cie.2003.03.006.

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Figure 1.  Batch-flow and one-piece flow
Figure 2.  Average cell response time (Solution decoding)
Figure 3.  Multi-Hybrid Cell Manufacturing System for the illustrative example
Figure 4.  Crossover operator
Figure 5.  Box-plot of RPD values with respect to the algorithms (for nine employees)
Figure 6.  Box-plots of RPD values obtained by the algorithms for each level of NBEC (for nine employees)
Figure 7.  Box-plots of RPD values obtained by the algorithms for each level of NESL (for nine employees)
Figure 8.  Box-plots of RPD values obtained by the ABC algorithm for each level of NESL (for nine employees)
Table 1.  Illustrative example data for the problem
$i$ $cx_i$ $cn_i$ $q_i$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $o_i$ $a_i$
166355515010051520164
286301002001555030164
3803051515550302010205
4552010101005020200205
$i$ $cx_i$ $cn_i$ $q_i$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $o_i$ $a_i$
166355515010051520164
286301002001555030164
3803051515550302010205
4552010101005020200205
Table 2.  Maximum and minimum number of employees
$i$ $W_i$ (Maximum) $W1_i$ (Minimum)
1 (Cell1)21
2 (Cell1)31
3 (Cell2)31
4 (Cell2)31
$i$ $W_i$ (Maximum) $W1_i$ (Minimum)
1 (Cell1)21
2 (Cell1)31
3 (Cell2)31
4 (Cell2)31
Table 3.  The batch list representation scheme (Encoding scheme)
position numbers1234
batch list2413
batch-employee assignment1-31-221-3
employee-machine assignment(1-2) (3-4)(2-3) (1-4)(1-2-3-4)(3-4) (1-2)
position numbers1234
batch list2413
batch-employee assignment1-31-221-3
employee-machine assignment(1-2) (3-4)(2-3) (1-4)(1-2-3-4)(3-4) (1-2)
Table 4.  The cell cycle times and the first lead times
$i$ $cn_i$employee1employee2employee3 $cy_i$ $FT_i$
1 (Cell1)35-66-6686
2 (Cell1)304343-4391
3 (Cell2)3050-2550120
4 (Cell2)203520-3585
$i$ $cn_i$employee1employee2employee3 $cy_i$ $FT_i$
1 (Cell1)35-66-6686
2 (Cell1)304343-4391
3 (Cell2)3050-2550120
4 (Cell2)203520-3585
Table 5.  Cell cycle times, batch sizes, operation times and total walking times
$i$ $cx_i$ $cn_i$ $q_i$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $e_{zim}$ $e_{zim}$$e_{zim}$ $o_i$
1957450516150MOMO4501551086216
2622468416150110358MO1601086216
39154321015144MOMO4321481055516
4590455515144100355MO1521055816
58874201515140MOMO420145965516
655543881514088350MO145965516
77413191512135MOMO319130844516
850835551213575280MO138844816
9626250109120MOMO250117714415
1043627210912060212MO117714415
11472155158108MOMO15588663215
123631869810826160MO102663815
13525182128115MOMO18295714014
144142108811550160MO112714414
156172261411131MOMO226102854814
1646925881113158200MO120855014
177553181014145MOMO318115945514
1854136051414580280MO134946014
19940431518153MOMO4311391186516
206534901518153120370MO1551187316
2110704851025168MOMO4851631357816
22739545825168135410MO1751358516
2312025281034185MOMO5281881619016
24857607534185147460MO21116110316
$i$ $cx_i$ $cn_i$ $q_i$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $e_{zim}$ $e_{zim}$$e_{zim}$ $o_i$
1957450516150MOMO4501551086216
2622468416150110358MO1601086216
39154321015144MOMO4321481055516
4590455515144100355MO1521055816
58874201515140MOMO420145965516
655543881514088350MO145965516
77413191512135MOMO319130844516
850835551213575280MO138844816
9626250109120MOMO250117714415
1043627210912060212MO117714415
11472155158108MOMO15588663215
123631869810826160MO102663815
13525182128115MOMO18295714014
144142108811550160MO112714414
156172261411131MOMO226102854814
1646925881113158200MO120855014
177553181014145MOMO318115945514
1854136051414580280MO134946014
19940431518153MOMO4311391186516
206534901518153120370MO1551187316
2110704851025168MOMO4851631357816
22739545825168135410MO1751358516
2312025281034185MOMO5281881619016
24857607534185147460MO21116110316
Table 6.  Experimental factors and their levels
FactorLevel      
Number of Batches on Each Cell (NBEC)1NB    
(Problem size factor)2[5$\times$ NB]    
  3[10$\times$ NB]    
Number of Employees for each Skill Level  JuniorNormalSenior
(NESL)133%33%33%
  266%33%00%
  366%00%33%
  4100%00%00%
  500%66%33%
  633%66%00%
  700%100%00%
  833%00%66%
  900%33%66%
  1000%00%100%
FactorLevel      
Number of Batches on Each Cell (NBEC)1NB    
(Problem size factor)2[5$\times$ NB]    
  3[10$\times$ NB]    
Number of Employees for each Skill Level  JuniorNormalSenior
(NESL)133%33%33%
  266%33%00%
  366%00%33%
  4100%00%00%
  500%66%33%
  633%66%00%
  700%100%00%
  833%00%66%
  900%33%66%
  1000%00%100%
Table 7.  The coefficient of skill levels
 JuniorNormalSenior
Skill level coefficients0.6311.29
 JuniorNormalSenior
Skill level coefficients0.6311.29
Table 8.  The promising values of the parameters for the metaheuristics
AlgorithmNotationValuesCombination
NBEC=1NBEC=2NBEC=3
GA $PS$20, 40, 60, 80,100406080
$pcross$0.2, 0.4, 0.6, 0.80.80.60.6
$pmutation$0.1, 0.2, 0.3, 0.40.30.20.2
ABC $NFS$20, 40, 60, 80,100404060
$\lambda$2, 4, 6, 8, 10244
$limit$2, 4, 6, 8, 10664
SA $initialtemp$ $10^3\times$(1, 3, 5, 7)135
$coolingrate$0.9, 0.95, 0.990.990.990.99
$epoch$5, 10, 15, 20101515
AlgorithmNotationValuesCombination
NBEC=1NBEC=2NBEC=3
GA $PS$20, 40, 60, 80,100406080
$pcross$0.2, 0.4, 0.6, 0.80.80.60.6
$pmutation$0.1, 0.2, 0.3, 0.40.30.20.2
ABC $NFS$20, 40, 60, 80,100404060
$\lambda$2, 4, 6, 8, 10244
$limit$2, 4, 6, 8, 10664
SA $initialtemp$ $10^3\times$(1, 3, 5, 7)135
$coolingrate$0.9, 0.95, 0.990.990.990.99
$epoch$5, 10, 15, 20101515
Table 9.  The tuned values of the parameters for the metaheuristics
AlgorithmNotationCombination
NBEC=1NBEC=2NBEC=3
GA $PS$30, 40, 50 50, 60, 7070, 80, 90
$pcross$ 0.7, 0.8, 0.90.5, 0.6, 0.70.5, 0.6, 0.7
$pmutation$ 0.25, 0.3, 0.35 0.15, 0.2, 0.250.15, 0.2, 0.25
ABC $NFS$ 30, 40, 5030, 40, 5050, 60, 70
$\lambda$1, 2, 33, 4, 53, 4, 5
$limit$5, 6, 75, 6, 73, 4, 5
SA $initialtemp$ 750, 1000,12502500,3000, 35004000,5000, 6000
$coolingrate$ 0.98, 0.990.98, 0.990.98, 0.99
$epoch$ 8, 10, 12 13, 15, 1713, 15, 17
AlgorithmNotationCombination
NBEC=1NBEC=2NBEC=3
GA $PS$30, 40, 50 50, 60, 7070, 80, 90
$pcross$ 0.7, 0.8, 0.90.5, 0.6, 0.70.5, 0.6, 0.7
$pmutation$ 0.25, 0.3, 0.35 0.15, 0.2, 0.250.15, 0.2, 0.25
ABC $NFS$ 30, 40, 5030, 40, 5050, 60, 70
$\lambda$1, 2, 33, 4, 53, 4, 5
$limit$5, 6, 75, 6, 73, 4, 5
SA $initialtemp$ 750, 1000,12502500,3000, 35004000,5000, 6000
$coolingrate$ 0.98, 0.990.98, 0.990.98, 0.99
$epoch$ 8, 10, 12 13, 15, 1713, 15, 17
Table 10.  The maximum computational times
ABCGASA
NBEC=114.8712.689.16
NBEC=253.4046.0738.73
NBEC=398.2595.5383.68
ABCGASA
NBEC=114.8712.689.16
NBEC=253.4046.0738.73
NBEC=398.2595.5383.68
Table 11.  Median of RPD values and computational time of algorithms for 9 employees
NESL$\times$NBECRPD
LBSAGAABCABCWLSGAWLSCPU
1$\times$12358017.6616.9618.821817, 1415
2$\times$11898919.3420.0116.0820.0421.3915
3$\times$12226317.9918.9716.6218.8319.0515
4$\times$12319620.2219.2317.418.5520.1315
5$\times$12222316.0517.6618.1918.9918.2515
6$\times$12209718.0519.1816.0217.3921.0515
7$\times$12068520.5718.8718.5318.3219.0815
8$\times$12435119.9321.0316.5617.3621.5815
9$\times$11962117.4518.618.5718.9220.415
10$\times$12396120.4220.5715.9720.0221.1615
Medians18.768 19.108 17.276 18.642 19.923 15
1$\times$212309334.9930.1326.8827.8532.1551.2
2$\times$211638333.2431.8129.8828.4432.7351.2
3$\times$212525928.8727.5529.1629.2829.7151.2
4$\times$211456430.3828.4928.5929.3629.7351.2
5$\times$210806528.3730.1629.7429.6532.7651.2
6$\times$212175030.8830.6729.7530.4932.651.2
7$\times$211927431.8531.9227.2131.9333.5851.2
8$\times$211857730.5629.6528.2432.129.1651.2
9$\times$210452129.5730.0325.6232.1230.2251.2
10$\times$211110029.4529.1129.6531.0335.5451.2
Medians30.816 29.952 28.472 30.225 31.818 51.2
1$\times$324239936.4535.1334.0536.4839.7296.9
2$\times$324924040.2839.5536.663641.1196.9
3$\times$323804539.7936.6134.8736.7639.9296.9
4$\times$324618737.2138.6235.0438.2439.5196.9
5$\times$322984236.4435.8736.4839.4641.0996.9
6$\times$322837536.3436.0136.1438.6339.4596.9
7$\times$321154037.3436.9135.9239.0438.8796.9
8$\times$322556240.2640.2736.2137.6141.7896.9
9$\times$323229939.7238.5333.3236.242.2196.9
10$\times$323751839.7637.634.7336.8641.0696.9
Medians38.359 37.51 35.342 37.528 40.472 96.9
NESL$\times$NBECRPD
LBSAGAABCABCWLSGAWLSCPU
1$\times$12358017.6616.9618.821817, 1415
2$\times$11898919.3420.0116.0820.0421.3915
3$\times$12226317.9918.9716.6218.8319.0515
4$\times$12319620.2219.2317.418.5520.1315
5$\times$12222316.0517.6618.1918.9918.2515
6$\times$12209718.0519.1816.0217.3921.0515
7$\times$12068520.5718.8718.5318.3219.0815
8$\times$12435119.9321.0316.5617.3621.5815
9$\times$11962117.4518.618.5718.9220.415
10$\times$12396120.4220.5715.9720.0221.1615
Medians18.768 19.108 17.276 18.642 19.923 15
1$\times$212309334.9930.1326.8827.8532.1551.2
2$\times$211638333.2431.8129.8828.4432.7351.2
3$\times$212525928.8727.5529.1629.2829.7151.2
4$\times$211456430.3828.4928.5929.3629.7351.2
5$\times$210806528.3730.1629.7429.6532.7651.2
6$\times$212175030.8830.6729.7530.4932.651.2
7$\times$211927431.8531.9227.2131.9333.5851.2
8$\times$211857730.5629.6528.2432.129.1651.2
9$\times$210452129.5730.0325.6232.1230.2251.2
10$\times$211110029.4529.1129.6531.0335.5451.2
Medians30.816 29.952 28.472 30.225 31.818 51.2
1$\times$324239936.4535.1334.0536.4839.7296.9
2$\times$324924040.2839.5536.663641.1196.9
3$\times$323804539.7936.6134.8736.7639.9296.9
4$\times$324618737.2138.6235.0438.2439.5196.9
5$\times$322984236.4435.8736.4839.4641.0996.9
6$\times$322837536.3436.0136.1438.6339.4596.9
7$\times$321154037.3436.9135.9239.0438.8796.9
8$\times$322556240.2640.2736.2137.6141.7896.9
9$\times$323229939.7238.5333.3236.242.2196.9
10$\times$323751839.7637.634.7336.8641.0696.9
Medians38.359 37.51 35.342 37.528 40.472 96.9
Table 12.  Median of RPD values and computational time of algorithms for 15 employees
NESL$\times$ NBECRPD
LBSAGAABCABCWLSGAWLSCPU
7$\times$1206853.683.453.113.533.6515
7$\times$21192745.114.94.655.075.1551.2
7$\times$32283756.325.985.65.926.4496.9
NESL$\times$ NBECRPD
LBSAGAABCABCWLSGAWLSCPU
7$\times$1206853.683.453.113.533.6515
7$\times$21192745.114.94.655.075.1551.2
7$\times$32283756.325.985.65.926.4496.9
Table 13.  Three-way ANOVA: RPD versus NBEC, NESL, and Algorithms
SourceFSig. ($p$)Partial eta squared
NBEC132.8020.0000.708
NESL1.1860.3950.155
Algorithms22.2280.0020.626
Interactions
NBEC*NESL1.3240.1990.362
NBEC*Algorithms1.2840.2660.226
NESL*Algorithms0.8140.7480.337
SourceFSig. ($p$)Partial eta squared
NBEC132.8020.0000.708
NESL1.1860.3950.155
Algorithms22.2280.0020.626
Interactions
NBEC*NESL1.3240.1990.362
NBEC*Algorithms1.2840.2660.226
NESL*Algorithms0.8140.7480.337
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