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

Omer Faruk Yilmaz; Mehmet Bulent Durmusoglu

doi:10.3934/jimo.2018007

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July 2018, 14(3): 1219-1249. 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

Omer Faruk Yilmaz ^1,, and Mehmet Bulent Durmusoglu ^2,

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 July 2018 Early access 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.

Keywords: Batch scheduling, skilled workforce assignment, dual-resource constrained, hybrid manufacturing cells, metaheuristics.

Mathematics Subject Classification: Primary: 90B35, 90C10; Secondary: 90C27.

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 and Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007

References:

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[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. Satoglu, M. 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.
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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. Sawyerr, B. 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. Azadeh, M. 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. Celano, A. 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. Cortes, J. Larraneta, L. Onieva, J. 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. Costa, F. 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. Costa, F. 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. Costa, S. 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. Daniels, B. 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. Davis, H. 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. Edis, C. 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. Egilmez, B. 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. Fahimnia, H. 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. Fan, M. 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. Fowler, P. 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. Li, X. 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. Liu, J. 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. Naso, M. Surico, B. 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. Niakan, A. Baboli, T. 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. Norman, W. Tharmmaphornphilas, K. L. Needy, B. 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. Ramesh, S. 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. Satoglu, M. 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. Schaller, J. 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. Sousa, T. Soares, H. Morais, R. 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.
[54]	G. A. Suer and R. R. Tummaluri, Multi-period operator assignment considering skills, learning and forgetting in labour-intensive cells, International Journal of Production Research, 46 (2008), 469-493. doi: 10.1080/00207540601138551.
[55]	R. Suri, Quick response manufacturing: A competitive strategy for the 21st century, In Proceedings of the 2002 POLCA Implementation Workshop, 141 (2002).
[56]	E. G Talbi, Metaheuristics: From Design to Implementation, John Wiley & Son Ltd., 2009. doi: 10.1002/9780470496916.
[57]	M. Unal, A. Ak, V. Topuz and H. Erdal, Optimization of PID Controllers Using ant Colony and Genetic Algorithms, Springer, Heidelberg, 2013.
[58]	S. Venkataramanaiah, Scheduling in cellular manufacturing systems: An heuristic approach, International Journal of Production Research, 46 (2008), 429-449. doi: 10.1080/00207540601138577.
[59]	J. X. Wang, Cellular Manufacturing: Mitigating Risk and Uncertainty, CRC Press, Boca Raton, USA, 2015. doi: 10.1201/b18009-1.
[60]	U. Wemmerlöv and N. L. Hyer, Cellular manufacturing in the US industry: A survey of users, International Journal of Production Research, 27 (1989), 1511-1530. doi: 10.1080/00207548908942637.
[61]	J. Xu, X. Xu and S. Q. Xie, Recent developments in Dual Resource Constrained (DRC) system research, European Journal of Operational Research, 215 (2011), 309-318. doi: 10.1016/j.ejor.2011.03.004.
[62]	O. F. Yilmaz, E. Cevikcan and M. B. Durmusoglu, Scheduling batches in multi hybrid cell manufacturing system considering worker resources: A case study from pipeline industry, Advances in Production Engineering & Management, 11 (2016), 192-206. doi: 10.14743/apem2016.3.220.
[63]	O. F. Yilmaz, B. Oztaysi, M. B. Durmusoglu and S. C. Oner, Determination of material handling equipment for lean in-plant logistics using fuzzy analytical network process considering risk attitudes of the experts, International Journal of Industrial Engineering: Theory, Applications and Practice, 24 (2017), 81-122.

Figure 1. Batch-flow and one-piece flow

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Figure 2. Average cell response time (Solution decoding)

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Figure 3. Multi-Hybrid Cell Manufacturing System for the illustrative example

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Figure 4. Crossover operator

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Figure 5. Box-plot of RPD values with respect to the algorithms (for nine employees)

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Figure 6. Box-plots of RPD values obtained by the algorithms for each level of NBEC (for nine employees)

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Figure 7. Box-plots of RPD values obtained by the algorithms for each level of NESL (for nine employees)

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Figure 8. Box-plots of RPD values obtained by the ABC algorithm for each level of NESL (for nine employees)

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Table 1. Illustrative example data for the problem

$i$	$cx_i$	$cn_i$	$q_i$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$o_i$	$a_i$
1	66	35	5	5	15	0	10	0	5	15	20	16	4
2	86	30	10	0	20	0	15	5	5	0	30	16	4
3	80	30	5	15	15	5	5	0	30	20	10	20	5
4	55	20	10	10	10	0	5	0	20	20	0	20	5

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$i$	$cx_i$	$cn_i$	$q_i$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$o_i$	$a_i$
1	66	35	5	5	15	0	10	0	5	15	20	16	4
2	86	30	10	0	20	0	15	5	5	0	30	16	4
3	80	30	5	15	15	5	5	0	30	20	10	20	5
4	55	20	10	10	10	0	5	0	20	20	0	20	5

Table 2. Maximum and minimum number of employees

$i$	$W_i$ (Maximum)	$W1_i$ (Minimum)
1 (Cell1)	2	1
2 (Cell1)	3	1
3 (Cell2)	3	1
4 (Cell2)	3	1

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$i$	$W_i$ (Maximum)	$W1_i$ (Minimum)
1 (Cell1)	2	1
2 (Cell1)	3	1
3 (Cell2)	3	1
4 (Cell2)	3	1

Table 3. The batch list representation scheme (Encoding scheme)

position numbers	1	2	3	4
batch list	2	4	1	3
batch-employee assignment	1-3	1-2	2	1-3
employee-machine assignment	(1-2) (3-4)	(2-3) (1-4)	(1-2-3-4)	(3-4) (1-2)

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position numbers	1	2	3	4
batch list	2	4	1	3
batch-employee assignment	1-3	1-2	2	1-3
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

$i$	$cn_i$	employee1	employee2	employee3	$cy_i$	$FT_i$
1 (Cell1)	35	-	66	-	66	86
2 (Cell1)	30	43	43	-	43	91
3 (Cell2)	30	50	-	25	50	120
4 (Cell2)	20	35	20	-	35	85

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$i$	$cn_i$	employee1	employee2	employee3	$cy_i$	$FT_i$
1 (Cell1)	35	-	66	-	66	86
2 (Cell1)	30	43	43	-	43	91
3 (Cell2)	30	50	-	25	50	120
4 (Cell2)	20	35	20	-	35	85

Table 5. Cell cycle times, batch sizes, operation times and total walking times

$i$	$cx_i$	$cn_i$	$q_i$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$o_i$
1	957	450	5	16	150	MO	MO	450	155	108	62	16
2	622	468	4	16	150	110	358	MO	160	108	62	16
3	915	432	10	15	144	MO	MO	432	148	105	55	16
4	590	455	5	15	144	100	355	MO	152	105	58	16
5	887	420	15	15	140	MO	MO	420	145	96	55	16
6	555	438	8	15	140	88	350	MO	145	96	55	16
7	741	319	15	12	135	MO	MO	319	130	84	45	16
8	508	355	5	12	135	75	280	MO	138	84	48	16
9	626	250	10	9	120	MO	MO	250	117	71	44	15
10	436	272	10	9	120	60	212	MO	117	71	44	15
11	472	155	15	8	108	MO	MO	155	88	66	32	15
12	363	186	9	8	108	26	160	MO	102	66	38	15
13	525	182	12	8	115	MO	MO	182	95	71	40	14
14	414	210	8	8	115	50	160	MO	112	71	44	14
15	617	226	14	11	131	MO	MO	226	102	85	48	14
16	469	258	8	11	131	58	200	MO	120	85	50	14
17	755	318	10	14	145	MO	MO	318	115	94	55	14
18	541	360	5	14	145	80	280	MO	134	94	60	14
19	940	431	5	18	153	MO	MO	431	139	118	65	16
20	653	490	15	18	153	120	370	MO	155	118	73	16
21	1070	485	10	25	168	MO	MO	485	163	135	78	16
22	739	545	8	25	168	135	410	MO	175	135	85	16
23	1202	528	10	34	185	MO	MO	528	188	161	90	16
24	857	607	5	34	185	147	460	MO	211	161	103	16

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$i$	$cx_i$	$cn_i$	$q_i$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$r_{im}$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$e_{zim}$	$o_i$
1	957	450	5	16	150	MO	MO	450	155	108	62	16
2	622	468	4	16	150	110	358	MO	160	108	62	16
3	915	432	10	15	144	MO	MO	432	148	105	55	16
4	590	455	5	15	144	100	355	MO	152	105	58	16
5	887	420	15	15	140	MO	MO	420	145	96	55	16
6	555	438	8	15	140	88	350	MO	145	96	55	16
7	741	319	15	12	135	MO	MO	319	130	84	45	16
8	508	355	5	12	135	75	280	MO	138	84	48	16
9	626	250	10	9	120	MO	MO	250	117	71	44	15
10	436	272	10	9	120	60	212	MO	117	71	44	15
11	472	155	15	8	108	MO	MO	155	88	66	32	15
12	363	186	9	8	108	26	160	MO	102	66	38	15
13	525	182	12	8	115	MO	MO	182	95	71	40	14
14	414	210	8	8	115	50	160	MO	112	71	44	14
15	617	226	14	11	131	MO	MO	226	102	85	48	14
16	469	258	8	11	131	58	200	MO	120	85	50	14
17	755	318	10	14	145	MO	MO	318	115	94	55	14
18	541	360	5	14	145	80	280	MO	134	94	60	14
19	940	431	5	18	153	MO	MO	431	139	118	65	16
20	653	490	15	18	153	120	370	MO	155	118	73	16
21	1070	485	10	25	168	MO	MO	485	163	135	78	16
22	739	545	8	25	168	135	410	MO	175	135	85	16
23	1202	528	10	34	185	MO	MO	528	188	161	90	16
24	857	607	5	34	185	147	460	MO	211	161	103	16

Table 6. Experimental factors and their levels

Factor	Level
Number of Batches on Each Cell (NBEC)	1	NB
(Problem size factor)	2	[5$\times$ NB]
	3	[10$\times$ NB]
Number of Employees for each Skill Level		Junior	Normal	Senior
(NESL)	1	33%	33%	33%
	2	66%	33%	00%
	3	66%	00%	33%
	4	100%	00%	00%
	5	00%	66%	33%
	6	33%	66%	00%
	7	00%	100%	00%
	8	33%	00%	66%
	9	00%	33%	66%
	10	00%	00%	100%

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Factor	Level
Number of Batches on Each Cell (NBEC)	1	NB
(Problem size factor)	2	[5$\times$ NB]
	3	[10$\times$ NB]
Number of Employees for each Skill Level		Junior	Normal	Senior
(NESL)	1	33%	33%	33%
	2	66%	33%	00%
	3	66%	00%	33%
	4	100%	00%	00%
	5	00%	66%	33%
	6	33%	66%	00%
	7	00%	100%	00%
	8	33%	00%	66%
	9	00%	33%	66%
	10	00%	00%	100%

Table 7. The coefficient of skill levels

	Junior	Normal	Senior
Skill level coefficients	0.63	1	1.29

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	Junior	Normal	Senior
Skill level coefficients	0.63	1	1.29

Table 8. The promising values of the parameters for the metaheuristics

Algorithm	Notation	Values		Combination
			NBEC=1	NBEC=2	NBEC=3
GA	$PS$	20, 40, 60, 80,100	40	60	80
	$pcross$	0.2, 0.4, 0.6, 0.8	0.8	0.6	0.6
	$pmutation$	0.1, 0.2, 0.3, 0.4	0.3	0.2	0.2
ABC	$NFS$	20, 40, 60, 80,100	40	40	60
	$\lambda$	2, 4, 6, 8, 10	2	4	4
	$limit$	2, 4, 6, 8, 10	6	6	4
SA	$initialtemp$	$10^3\times$(1, 3, 5, 7)	1	3	5
	$coolingrate$	0.9, 0.95, 0.99	0.99	0.99	0.99
	$epoch$	5, 10, 15, 20	10	15	15

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Algorithm	Notation	Values		Combination
			NBEC=1	NBEC=2	NBEC=3
GA	$PS$	20, 40, 60, 80,100	40	60	80
	$pcross$	0.2, 0.4, 0.6, 0.8	0.8	0.6	0.6
	$pmutation$	0.1, 0.2, 0.3, 0.4	0.3	0.2	0.2
ABC	$NFS$	20, 40, 60, 80,100	40	40	60
	$\lambda$	2, 4, 6, 8, 10	2	4	4
	$limit$	2, 4, 6, 8, 10	6	6	4
SA	$initialtemp$	$10^3\times$(1, 3, 5, 7)	1	3	5
	$coolingrate$	0.9, 0.95, 0.99	0.99	0.99	0.99
	$epoch$	5, 10, 15, 20	10	15	15

Table 9. The tuned values of the parameters for the metaheuristics

Algorithm	Notation	Combination
		NBEC=1	NBEC=2	NBEC=3
GA	$PS$	30, 40, 50	50, 60, 70	70, 80, 90
	$pcross$	0.7, 0.8, 0.9	0.5, 0.6, 0.7	0.5, 0.6, 0.7
	$pmutation$	0.25, 0.3, 0.35	0.15, 0.2, 0.25	0.15, 0.2, 0.25
ABC	$NFS$	30, 40, 50	30, 40, 50	50, 60, 70
	$\lambda$	1, 2, 3	3, 4, 5	3, 4, 5
	$limit$	5, 6, 7	5, 6, 7	3, 4, 5
SA	$initialtemp$	750, 1000,1250	2500,3000, 3500	4000,5000, 6000
	$coolingrate$	0.98, 0.99	0.98, 0.99	0.98, 0.99
	$epoch$	8, 10, 12	13, 15, 17	13, 15, 17

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Algorithm	Notation	Combination
		NBEC=1	NBEC=2	NBEC=3
GA	$PS$	30, 40, 50	50, 60, 70	70, 80, 90
	$pcross$	0.7, 0.8, 0.9	0.5, 0.6, 0.7	0.5, 0.6, 0.7
	$pmutation$	0.25, 0.3, 0.35	0.15, 0.2, 0.25	0.15, 0.2, 0.25
ABC	$NFS$	30, 40, 50	30, 40, 50	50, 60, 70
	$\lambda$	1, 2, 3	3, 4, 5	3, 4, 5
	$limit$	5, 6, 7	5, 6, 7	3, 4, 5
SA	$initialtemp$	750, 1000,1250	2500,3000, 3500	4000,5000, 6000
	$coolingrate$	0.98, 0.99	0.98, 0.99	0.98, 0.99
	$epoch$	8, 10, 12	13, 15, 17	13, 15, 17

Table 10. The maximum computational times

	ABC	GA	SA
NBEC=1	14.87	12.68	9.16
NBEC=2	53.40	46.07	38.73
NBEC=3	98.25	95.53	83.68

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	ABC	GA	SA
NBEC=1	14.87	12.68	9.16
NBEC=2	53.40	46.07	38.73
NBEC=3	98.25	95.53	83.68

Table 11. Median of RPD values and computational time of algorithms for 9 employees

NESL$\times$NBEC				RPD
NESL$\times$NBEC	LB	SA	GA	ABC	ABCWLS	GAWLS	CPU
1$\times$1	23580	17.66	16.96	18.82	18	17, 14	15
2$\times$1	18989	19.34	20.01	16.08	20.04	21.39	15
3$\times$1	22263	17.99	18.97	16.62	18.83	19.05	15
4$\times$1	23196	20.22	19.23	17.4	18.55	20.13	15
5$\times$1	22223	16.05	17.66	18.19	18.99	18.25	15
6$\times$1	22097	18.05	19.18	16.02	17.39	21.05	15
7$\times$1	20685	20.57	18.87	18.53	18.32	19.08	15
8$\times$1	24351	19.93	21.03	16.56	17.36	21.58	15
9$\times$1	19621	17.45	18.6	18.57	18.92	20.4	15
10$\times$1	23961	20.42	20.57	15.97	20.02	21.16	15
Medians		18.768	19.108	17.276	18.642	19.923	15
1$\times$2	123093	34.99	30.13	26.88	27.85	32.15	51.2
2$\times$2	116383	33.24	31.81	29.88	28.44	32.73	51.2
3$\times$2	125259	28.87	27.55	29.16	29.28	29.71	51.2
4$\times$2	114564	30.38	28.49	28.59	29.36	29.73	51.2
5$\times$2	108065	28.37	30.16	29.74	29.65	32.76	51.2
6$\times$2	121750	30.88	30.67	29.75	30.49	32.6	51.2
7$\times$2	119274	31.85	31.92	27.21	31.93	33.58	51.2
8$\times$2	118577	30.56	29.65	28.24	32.1	29.16	51.2
9$\times$2	104521	29.57	30.03	25.62	32.12	30.22	51.2
10$\times$2	111100	29.45	29.11	29.65	31.03	35.54	51.2
Medians		30.816	29.952	28.472	30.225	31.818	51.2
1$\times$3	242399	36.45	35.13	34.05	36.48	39.72	96.9
2$\times$3	249240	40.28	39.55	36.66	36	41.11	96.9
3$\times$3	238045	39.79	36.61	34.87	36.76	39.92	96.9
4$\times$3	246187	37.21	38.62	35.04	38.24	39.51	96.9
5$\times$3	229842	36.44	35.87	36.48	39.46	41.09	96.9
6$\times$3	228375	36.34	36.01	36.14	38.63	39.45	96.9
7$\times$3	211540	37.34	36.91	35.92	39.04	38.87	96.9
8$\times$3	225562	40.26	40.27	36.21	37.61	41.78	96.9
9$\times$3	232299	39.72	38.53	33.32	36.2	42.21	96.9
10$\times$3	237518	39.76	37.6	34.73	36.86	41.06	96.9
Medians		38.359	37.51	35.342	37.528	40.472	96.9

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NESL$\times$NBEC				RPD
NESL$\times$NBEC	LB	SA	GA	ABC	ABCWLS	GAWLS	CPU
1$\times$1	23580	17.66	16.96	18.82	18	17, 14	15
2$\times$1	18989	19.34	20.01	16.08	20.04	21.39	15
3$\times$1	22263	17.99	18.97	16.62	18.83	19.05	15
4$\times$1	23196	20.22	19.23	17.4	18.55	20.13	15
5$\times$1	22223	16.05	17.66	18.19	18.99	18.25	15
6$\times$1	22097	18.05	19.18	16.02	17.39	21.05	15
7$\times$1	20685	20.57	18.87	18.53	18.32	19.08	15
8$\times$1	24351	19.93	21.03	16.56	17.36	21.58	15
9$\times$1	19621	17.45	18.6	18.57	18.92	20.4	15
10$\times$1	23961	20.42	20.57	15.97	20.02	21.16	15
Medians		18.768	19.108	17.276	18.642	19.923	15
1$\times$2	123093	34.99	30.13	26.88	27.85	32.15	51.2
2$\times$2	116383	33.24	31.81	29.88	28.44	32.73	51.2
3$\times$2	125259	28.87	27.55	29.16	29.28	29.71	51.2
4$\times$2	114564	30.38	28.49	28.59	29.36	29.73	51.2
5$\times$2	108065	28.37	30.16	29.74	29.65	32.76	51.2
6$\times$2	121750	30.88	30.67	29.75	30.49	32.6	51.2
7$\times$2	119274	31.85	31.92	27.21	31.93	33.58	51.2
8$\times$2	118577	30.56	29.65	28.24	32.1	29.16	51.2
9$\times$2	104521	29.57	30.03	25.62	32.12	30.22	51.2
10$\times$2	111100	29.45	29.11	29.65	31.03	35.54	51.2
Medians		30.816	29.952	28.472	30.225	31.818	51.2
1$\times$3	242399	36.45	35.13	34.05	36.48	39.72	96.9
2$\times$3	249240	40.28	39.55	36.66	36	41.11	96.9
3$\times$3	238045	39.79	36.61	34.87	36.76	39.92	96.9
4$\times$3	246187	37.21	38.62	35.04	38.24	39.51	96.9
5$\times$3	229842	36.44	35.87	36.48	39.46	41.09	96.9
6$\times$3	228375	36.34	36.01	36.14	38.63	39.45	96.9
7$\times$3	211540	37.34	36.91	35.92	39.04	38.87	96.9
8$\times$3	225562	40.26	40.27	36.21	37.61	41.78	96.9
9$\times$3	232299	39.72	38.53	33.32	36.2	42.21	96.9
10$\times$3	237518	39.76	37.6	34.73	36.86	41.06	96.9
Medians		38.359	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$\times$ NBEC				RPD
NESL$\times$ NBEC	LB	SA	GA	ABC	ABCWLS	GAWLS	CPU
7$\times$1	20685	3.68	3.45	3.11	3.53	3.65	15
7$\times$2	119274	5.11	4.9	4.65	5.07	5.15	51.2
7$\times$3	228375	6.32	5.98	5.6	5.92	6.44	96.9

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NESL$\times$ NBEC				RPD
NESL$\times$ NBEC	LB	SA	GA	ABC	ABCWLS	GAWLS	CPU
7$\times$1	20685	3.68	3.45	3.11	3.53	3.65	15
7$\times$2	119274	5.11	4.9	4.65	5.07	5.15	51.2
7$\times$3	228375	6.32	5.98	5.6	5.92	6.44	96.9

Table 13. Three-way ANOVA: RPD versus NBEC, NESL, and Algorithms

Source	F	Sig. ($p$)	Partial eta squared
NBEC	132.802	0.000	0.708
NESL	1.186	0.395	0.155
Algorithms	22.228	0.002	0.626
Interactions
NBEC*NESL	1.324	0.199	0.362
NBEC*Algorithms	1.284	0.266	0.226
NESL*Algorithms	0.814	0.748	0.337

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Source	F	Sig. ($p$)	Partial eta squared
NBEC	132.802	0.000	0.708
NESL	1.186	0.395	0.155
Algorithms	22.228	0.002	0.626
Interactions
NBEC*NESL	1.324	0.199	0.362
NBEC*Algorithms	1.284	0.266	0.226
NESL*Algorithms	0.814	0.748	0.337

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2021 Impact Factor: 1.411

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2021 Impact Factor: 1.411

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2021 Impact Factor: 1.411

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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