An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm

Behrad Erfani; Sadoullah Ebrahimnejad; Amirhossein Moosavi

doi:10.3934/jimo.2019030

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July 2020, 16(4): 1801-1834. doi: 10.3934/jimo.2019030

An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm

Behrad Erfani , Sadoullah Ebrahimnejad ^, and Amirhossein Moosavi

Department of Industrial Engineering, Karaj Branch, Islamic Azad University, Karaj, Iran

Corresponding author: Sadoullah Ebrahimnejad^*

Received February 2018 Revised October 2018 Published July 2020 Early access May 2019

The aim of this research is to study the dynamic facility layout and job-shop scheduling problems, simultaneously. In fact, this paper intends to measure the synergy between these two problems. In this paper, a multi-objective mixed integer nonlinear programming model has been proposed where areas of departments are unequal. Using a new approach, this paper calculates the farness rating scores of departments beside their closeness rating scores. Another feature of this paper is the consideration of input and output points for each department, which is crucial for the establishment of practical facility layouts in the real world. In the scheduling problem, transportation delay between departments and machines' setup time are considered that affect the dynamic facility layout problem. This integrated problem is solved using a hybrid two-phase algorithm. In the first phase, this hybrid algorithm incorporates the non-dominated sorting genetic algorithm. The second phase also applies two local search algorithms. To increase the efficacy of the first phase, we have tuned the parameters of this phase using the Taguchi experimental design method. Then, we have randomly generated 20 instances of different sizes. The numerical results show that the second phase of the hybrid algorithm improves its first phase significantly. The results also demonstrate that the simultaneous optimization of those two problems decreases the mean flow time of jobs by about 10% as compared to their separate optimization.

Keywords: Dynamic facility layout planning, job shop scheduling, multi-objective optimization, unequal area, NSGA-II, local search algorithm.

Mathematics Subject Classification: Primary: 90B50; Secondary: 90B85, 90B35.

Citation: Behrad Erfani, Sadoullah Ebrahimnejad, Amirhossein Moosavi. An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1801-1834. doi: 10.3934/jimo.2019030

References:

[1]	A. D. Asl and K. Y. Wong, Solving unequal-area static and dynamic facility layout problems using modified particle swarm optimization, Journal of Intelligent Manufacturing, 28 (2017), 1317-1336. doi: 10.1007/s10845-015-1053-5. Google Scholar
[2]	C. Bierwirth and J. Kuhpfahl, Extended GRASP for the job shop scheduling problem with total weighted tardiness objective, European Journal of Operational Research, 261 (2017), 835-848. doi: 10.1016/j.ejor.2017.03.030. Google Scholar
[3]	H. X. Chen and H. C. Lau, A math-heuristic approach for integrated resource scheduling in a maritime logistics facility, 2011 IEEE International Conference on Industrial Engineering and Engineering Management, (2011). doi: 10.1109/IEEM.2011.6117906. Google Scholar
[4]	K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE transactions on Evolutionary Computation, 6 (2002), 182-197. doi: 10.1109/4235.996017. Google Scholar
[5]	S. Emami and A. S. Nookabadi, Managing a new multi-objective model for the dynamic facility layout problem, The International Journal of Advanced Manufacturing Technology, 68 (2013), 2215-2228. doi: 10.1007/s00170-013-4820-5. Google Scholar
[6]	X. Hao, M. Gen, L. Lin and G. A. Suer, Effective multiobjective EDA for bi-criteria stochastic job-shop scheduling problem, Journal of Intelligent Manufacturing, 28 (2017), 833-845. doi: 10.1007/s10845-014-1026-0. Google Scholar
[7]	M. Kaveh, V. M. Dalfard and S. Amiri, A new intelligent algorithm for dynamic facility layout problem in state of fuzzy constraints, Neural Computing and Applications, 24 (2014), 1179-1190. doi: 10.1007/s00521-013-1339-5. Google Scholar
[8]	N. Khilwani, R. Shankar and M. Tiwari, Facility layout problem: An approach based on a group decision-making system and psychoclonal algorithm, International Journal of Production Research, 46 (2008), 895-927. doi: 10.1080/00207540600943993. Google Scholar
[9]	R. Kolisch, A. Sprecher and A. Drexl, Characterization and generation of a general class of resource-constrained project scheduling problems, Management Science, 41 (1995), 1693-1703. doi: 10.1287/mnsc.41.10.1693. Google Scholar
[10]	J. Liu, D. Wang, K. He and Y. Xue, Combining Wang-Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem, European Journal of Operational Research, 262 (2017), 1052-1063. doi: 10.1016/j.ejor.2017.04.002. Google Scholar
[11]	G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Applied Mathematics and Computation, 213 (2009), 455-465. doi: 10.1016/j.amc.2009.03.037. Google Scholar
[12]	A. R. McKendall and A. Hakobyan, Heuristics for the dynamic facility layout problem with unequal-area departments, European Journal of Operational Research, 201 (2010), 171-182. doi: 10.1016/j.ejor.2009.02.028. Google Scholar
[13]	N. Mladenović and P. Hansen, Variable neighborhood search, Computers & Operations Research, 11 (1997), 1097-1100. doi: 10.1016/S0305-0548(97)00031-2. Google Scholar
[14]	A. Mohamadi, S. Ebrahimnejad and R. Tavakkoli-Moghaddam, A novel two-stage approach for solving a bi-objective facility layout problem, International Journal of Operational Research, 31 (2018), 49-87. doi: 10.1504/IJOR.2018.088557. Google Scholar
[15]	A. Moosavi and S. Ebrahimnejad, Scheduling of elective patients considering upstream and downstream units and emergency demand using robust optimization, Computers & Industrial Engineering, 120 (2018), 216-233. doi: 10.1016/j.cie.2018.04.047. Google Scholar
[16]	N. Nekooghadirli, R. Tavakkoli-Moghaddam, V. R. Ghezavati and S. Javanmard, Solving a new bi-objective location-routing-inventory problem in a distribution network by meta-heuristics, Computers & Industrial Engineering, 76 (2014), 204-221. doi: 10.1016/j.cie.2014.08.004. Google Scholar
[17]	M. Pirayesh and S. Poormoaied, Location and job shop scheduling problem in fuzzy environment, 5th Int. Conference of the Iranian Society of Operations Research, Azarbaijan, Iran, (2012). Google Scholar
[18]	H. Pourvaziri and H. Pierreval, Dynamic facility layout problem based on open queuing network theory, European Journal of Operational Research, 259 (2017), 538-553. doi: 10.1016/j.ejor.2016.11.011. Google Scholar
[19]	K. S. N. Ripon and J. Torresen, Integrated job shop scheduling and layout planning: a hybrid evolutionary method for optimizing multiple objectives, Evolving Systems, 5 (2014), 121-132. doi: 10.1007/s12530-013-9092-7. Google Scholar
[20]	K. S. N. Ripon, C. H. Tsang and S. Kwong, An evolutionary approach for solving the multi-objective job-shop scheduling problem, In Evolutionary Scheduling, 2007,165–195, Springer, Berlin, Heidelberg. doi: 10.1007/978-3-540-48584-1_7. Google Scholar
[21]	M. H. Salmani, K. Eshghi and H. Neghabi, A bi-objective MIP model for facility layout problem in uncertain environment, The International Journal of Advanced Manufacturing Technology, 81 (2015), 1563-1575. doi: 10.1007/s00170-015-7290-0. Google Scholar
[22]	H. Samarghandi, P. Taabayan and M. Behroozi, Metaheuristics for fuzzy dynamic facility layout problem with unequal area constraints and closeness ratings, The International Journal of Advanced Manufacturing Technology, 67 (2013), 2701-2715. doi: 10.1007/s00170-012-4685-z. Google Scholar
[23]	J. Shahrabi, M. A. Adibi and M. Mahootchi, A reinforcement learning approach to parameter estimation in dynamic job shop scheduling, Computers & Industrial Engineering, 110 (2017), 75-82. doi: 10.1016/j.cie.2017.05.026. Google Scholar
[24]	A. Srinivasan, Integrating Block Layout Design and Location of Input and Output Points in Facility Layout Problems, M.Sc. thesis, Concordia University in Canada, 2014. Google Scholar
[25]	C. R. Vela, R. Varela and M. A. González, Local search and genetic algorithm for the job shop scheduling problem with sequence dependent setup times, Journal of Heuristics, 16 (2010), 139-165. doi: 10.1007/s10732-008-9094-y. Google Scholar
[26]	L. Wang, Combining facility layout redesign and dynamic routing for job-shop assembly operations, 2011 IEEE International Symposium on Assembly and Manufacturing, Tampere, Finland, (2011). doi: 10.1109/ISAM.2011.5942302. Google Scholar
[27]	L. Wang, S. Keshavarzmanesh and H. Y. Feng, A hybrid approach for dynamic assembly shop floor layout, 2010 IEEE International Conference on Automation Science and Engineering, Toronto, ON, Canada, (2010). doi: 10.1109/COASE.2010.5584219. Google Scholar
[28]	L. Wang, H. Wu, F. Tang and D. Z. Zheng, A hybrid quantum-inspired genetic algorithm for flow shop scheduling, International Conference on Intelligent Computing, Berlin, Heidelberg, (2005), 636–644. doi: 10.1007/11538356_66. Google Scholar
[29]	C. L. Yang, S. P. Chuang and T. S. Hsu, A genetic algorithm for dynamic facility planning in job shop manufacturing, The International Journal of Advanced Manufacturing Technology, 52 (2011), 303-309. doi: 10.1007/s00170-010-2733-0. Google Scholar

show all references

References:

[1]	A. D. Asl and K. Y. Wong, Solving unequal-area static and dynamic facility layout problems using modified particle swarm optimization, Journal of Intelligent Manufacturing, 28 (2017), 1317-1336. doi: 10.1007/s10845-015-1053-5. Google Scholar
[2]	C. Bierwirth and J. Kuhpfahl, Extended GRASP for the job shop scheduling problem with total weighted tardiness objective, European Journal of Operational Research, 261 (2017), 835-848. doi: 10.1016/j.ejor.2017.03.030. Google Scholar
[3]	H. X. Chen and H. C. Lau, A math-heuristic approach for integrated resource scheduling in a maritime logistics facility, 2011 IEEE International Conference on Industrial Engineering and Engineering Management, (2011). doi: 10.1109/IEEM.2011.6117906. Google Scholar
[4]	K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE transactions on Evolutionary Computation, 6 (2002), 182-197. doi: 10.1109/4235.996017. Google Scholar
[5]	S. Emami and A. S. Nookabadi, Managing a new multi-objective model for the dynamic facility layout problem, The International Journal of Advanced Manufacturing Technology, 68 (2013), 2215-2228. doi: 10.1007/s00170-013-4820-5. Google Scholar
[6]	X. Hao, M. Gen, L. Lin and G. A. Suer, Effective multiobjective EDA for bi-criteria stochastic job-shop scheduling problem, Journal of Intelligent Manufacturing, 28 (2017), 833-845. doi: 10.1007/s10845-014-1026-0. Google Scholar
[7]	M. Kaveh, V. M. Dalfard and S. Amiri, A new intelligent algorithm for dynamic facility layout problem in state of fuzzy constraints, Neural Computing and Applications, 24 (2014), 1179-1190. doi: 10.1007/s00521-013-1339-5. Google Scholar
[8]	N. Khilwani, R. Shankar and M. Tiwari, Facility layout problem: An approach based on a group decision-making system and psychoclonal algorithm, International Journal of Production Research, 46 (2008), 895-927. doi: 10.1080/00207540600943993. Google Scholar
[9]	R. Kolisch, A. Sprecher and A. Drexl, Characterization and generation of a general class of resource-constrained project scheduling problems, Management Science, 41 (1995), 1693-1703. doi: 10.1287/mnsc.41.10.1693. Google Scholar
[10]	J. Liu, D. Wang, K. He and Y. Xue, Combining Wang-Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem, European Journal of Operational Research, 262 (2017), 1052-1063. doi: 10.1016/j.ejor.2017.04.002. Google Scholar
[11]	G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Applied Mathematics and Computation, 213 (2009), 455-465. doi: 10.1016/j.amc.2009.03.037. Google Scholar
[12]	A. R. McKendall and A. Hakobyan, Heuristics for the dynamic facility layout problem with unequal-area departments, European Journal of Operational Research, 201 (2010), 171-182. doi: 10.1016/j.ejor.2009.02.028. Google Scholar
[13]	N. Mladenović and P. Hansen, Variable neighborhood search, Computers & Operations Research, 11 (1997), 1097-1100. doi: 10.1016/S0305-0548(97)00031-2. Google Scholar
[14]	A. Mohamadi, S. Ebrahimnejad and R. Tavakkoli-Moghaddam, A novel two-stage approach for solving a bi-objective facility layout problem, International Journal of Operational Research, 31 (2018), 49-87. doi: 10.1504/IJOR.2018.088557. Google Scholar
[15]	A. Moosavi and S. Ebrahimnejad, Scheduling of elective patients considering upstream and downstream units and emergency demand using robust optimization, Computers & Industrial Engineering, 120 (2018), 216-233. doi: 10.1016/j.cie.2018.04.047. Google Scholar
[16]	N. Nekooghadirli, R. Tavakkoli-Moghaddam, V. R. Ghezavati and S. Javanmard, Solving a new bi-objective location-routing-inventory problem in a distribution network by meta-heuristics, Computers & Industrial Engineering, 76 (2014), 204-221. doi: 10.1016/j.cie.2014.08.004. Google Scholar
[17]	M. Pirayesh and S. Poormoaied, Location and job shop scheduling problem in fuzzy environment, 5th Int. Conference of the Iranian Society of Operations Research, Azarbaijan, Iran, (2012). Google Scholar
[18]	H. Pourvaziri and H. Pierreval, Dynamic facility layout problem based on open queuing network theory, European Journal of Operational Research, 259 (2017), 538-553. doi: 10.1016/j.ejor.2016.11.011. Google Scholar
[19]	K. S. N. Ripon and J. Torresen, Integrated job shop scheduling and layout planning: a hybrid evolutionary method for optimizing multiple objectives, Evolving Systems, 5 (2014), 121-132. doi: 10.1007/s12530-013-9092-7. Google Scholar
[20]	K. S. N. Ripon, C. H. Tsang and S. Kwong, An evolutionary approach for solving the multi-objective job-shop scheduling problem, In Evolutionary Scheduling, 2007,165–195, Springer, Berlin, Heidelberg. doi: 10.1007/978-3-540-48584-1_7. Google Scholar
[21]	M. H. Salmani, K. Eshghi and H. Neghabi, A bi-objective MIP model for facility layout problem in uncertain environment, The International Journal of Advanced Manufacturing Technology, 81 (2015), 1563-1575. doi: 10.1007/s00170-015-7290-0. Google Scholar
[22]	H. Samarghandi, P. Taabayan and M. Behroozi, Metaheuristics for fuzzy dynamic facility layout problem with unequal area constraints and closeness ratings, The International Journal of Advanced Manufacturing Technology, 67 (2013), 2701-2715. doi: 10.1007/s00170-012-4685-z. Google Scholar
[23]	J. Shahrabi, M. A. Adibi and M. Mahootchi, A reinforcement learning approach to parameter estimation in dynamic job shop scheduling, Computers & Industrial Engineering, 110 (2017), 75-82. doi: 10.1016/j.cie.2017.05.026. Google Scholar
[24]	A. Srinivasan, Integrating Block Layout Design and Location of Input and Output Points in Facility Layout Problems, M.Sc. thesis, Concordia University in Canada, 2014. Google Scholar
[25]	C. R. Vela, R. Varela and M. A. González, Local search and genetic algorithm for the job shop scheduling problem with sequence dependent setup times, Journal of Heuristics, 16 (2010), 139-165. doi: 10.1007/s10732-008-9094-y. Google Scholar
[26]	L. Wang, Combining facility layout redesign and dynamic routing for job-shop assembly operations, 2011 IEEE International Symposium on Assembly and Manufacturing, Tampere, Finland, (2011). doi: 10.1109/ISAM.2011.5942302. Google Scholar
[27]	L. Wang, S. Keshavarzmanesh and H. Y. Feng, A hybrid approach for dynamic assembly shop floor layout, 2010 IEEE International Conference on Automation Science and Engineering, Toronto, ON, Canada, (2010). doi: 10.1109/COASE.2010.5584219. Google Scholar
[28]	L. Wang, H. Wu, F. Tang and D. Z. Zheng, A hybrid quantum-inspired genetic algorithm for flow shop scheduling, International Conference on Intelligent Computing, Berlin, Heidelberg, (2005), 636–644. doi: 10.1007/11538356_66. Google Scholar
[29]	C. L. Yang, S. P. Chuang and T. S. Hsu, A genetic algorithm for dynamic facility planning in job shop manufacturing, The International Journal of Advanced Manufacturing Technology, 52 (2011), 303-309. doi: 10.1007/s00170-010-2733-0. Google Scholar

Figure 1. Two possible cases that could happen to determine the start time of a job

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Figure 2. An illustrative example of a solution with 12 departments and 20 jobs

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Figure 3. The candidate locations and departments arrangements for an example with 12 departments

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Figure 4. An illustrative example of the calculation of PUS

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Figure 5. The possible movements, and rotations in the local search for layout

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Figure 6. The flowchart of the second phase of the hybrid algorithm (local search algorithms)

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Figure 7. Illustrative examples of the violation of departments

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Figure 8. Illustration of the solution found for the discrete facility layout of Instance 15

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Figure 9. Illustration of the solution found for the continuous facility layout of Instance 15

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Figure 10. Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 1)

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Figure 11. Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 2)

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Figure 12. The distribution and the interval estimation of the assessment metrics for separate optimization and simultaneous optimization

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Table 1. The features and objectives studied in the literature

Problem	Rows	Features	Rows	Objectives
FLP	[F1]	Inequality of departments	[O1]	Material handling cost
	[F2]	Input and output for departments	[O2]	Rearrangement cost of departments
	[F3]	Multiple periods	[O3]	Desirability of closeness rating scores
	[F4]	Continuous Optimization	[O4]	PUS
			[O5]	Work in process
JSS	[F5]	Setup time	[O6]	Makespan
	[F6]	Transportation delay time	[O7]	Mean Flow Time (MFT)
	[F7]	Multiple periods	[O8]	Earliness
	[F8]	Due date of jobs	[O9]	Lateness
	[F9]	Machine breakdown

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Problem	Rows	Features	Rows	Objectives
FLP	[F1]	Inequality of departments	[O1]	Material handling cost
	[F2]	Input and output for departments	[O2]	Rearrangement cost of departments
	[F3]	Multiple periods	[O3]	Desirability of closeness rating scores
	[F4]	Continuous Optimization	[O4]	PUS
			[O5]	Work in process
JSS	[F5]	Setup time	[O6]	Makespan
	[F6]	Transportation delay time	[O7]	Mean Flow Time (MFT)
	[F7]	Multiple periods	[O8]	Earliness
	[F8]	Due date of jobs	[O9]	Lateness
	[F9]	Machine breakdown

Table 2. A summary of the features for a number of studies published recently

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Table 3. Specifications of randomly generated instances

Size of	Instance	No. of	No. of	No. of
instances	(No. of periods)	departments	machines	jobs
Small	1 (2), 11 (3)	3	3	3
	2 (2), 12 (3)	4	5	5
	3 (2), 13 (3)	5	7	7
Medium	4 (2), 14 (3)	6	9	9
	5 (2), 15 (3)	8	11	11
	6 (2), 16 (3)	10	13	13
Large-scale	7 (2), 17 (3)	12	16	16
	8 (2), 18 (3)	14	19	19
	9 (2), 19 (3)	16	21	21
	10 (2), 20 (3)	18	23	23

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Size of	Instance	No. of	No. of	No. of
instances	(No. of periods)	departments	machines	jobs
Small	1 (2), 11 (3)	3	3	3
	2 (2), 12 (3)	4	5	5
	3 (2), 13 (3)	5	7	7
Medium	4 (2), 14 (3)	6	9	9
	5 (2), 15 (3)	8	11	11
	6 (2), 16 (3)	10	13	13
Large-scale	7 (2), 17 (3)	12	16	16
	8 (2), 18 (3)	14	19	19
	9 (2), 19 (3)	16	21	21
	10 (2), 20 (3)	18	23	23

Table 4. The demand for products over different periods ([7])

	1 (*10)	2 (*10)	3 (*10)
1	T(250,280,300)	T(40, 50, 60)	T(40, 50, 60)
2	T(70, 75, 90)	T(350,400,430)	T(110,125,135)
3	N(5, 56)	N(2, 55)	N(20,550)
4	N(4, 40)	N(4, 50)	N(4, 70)

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	1 (*10)	2 (*10)	3 (*10)
1	T(250,280,300)	T(40, 50, 60)	T(40, 50, 60)
2	T(70, 75, 90)	T(350,400,430)	T(110,125,135)
3	N(5, 56)	N(2, 55)	N(20,550)
4	N(4, 40)	N(4, 50)	N(4, 70)

Table 5. The levels of parameters defined for experiments

Parameter	Level of parameters
	Small size			Medium size			Large-scale
	I	II	III	I	II	III	I	II	III
Iteration	60	80	100	80	100	120	100	150	200
Initial population	10	20	30	30	40	50	80	100	120
$ C_p $	0.7	0.8	0.9	0.7	0.8	0.9	0.7	0.8	0.9
$ M_p $	0.1	0.2	0.3	0.1	0.2	0.3	0.1	0.2	0.3

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Parameter	Level of parameters
	Small size			Medium size			Large-scale
	I	II	III	I	II	III	I	II	III
Iteration	60	80	100	80	100	120	100	150	200
Initial population	10	20	30	30	40	50	80	100	120
$ C_p $	0.7	0.8	0.9	0.7	0.8	0.9	0.7	0.8	0.9
$ M_p $	0.1	0.2	0.3	0.1	0.2	0.3	0.1	0.2	0.3

Table 6. The optimal setting for the parameters of NSGA-II

Parameter	Size of instances
	Small	Medium	Large-scale
Iteration	60	100	150
Initial population	20	30	100
$ C_p $	0.7	0.7	0.8
$ M_p $	0.2	0.3	0.3

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Parameter	Size of instances
	Small	Medium	Large-scale
Iteration	60	100	150
Initial population	20	30	100
$ C_p $	0.7	0.7	0.8
$ M_p $	0.2	0.3	0.3

Table 7. The comparison of the traditional and proposed method of the PUS

	4	7	10	13	16	19
Traditional method $ (\%) $	35.2	48.8	57.3	34	55.6	51.6
Proposed method $ (\%) $	38.9	41.1	49.8	35.3	42	37.6
Gap $ (\%) $	-10.6	15.9	13.1	-3.8	24.3	27

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	4	7	10	13	16	19
Traditional method $ (\%) $	35.2	48.8	57.3	34	55.6	51.6
Proposed method $ (\%) $	38.9	41.1	49.8	35.3	42	37.6
Gap $ (\%) $	-10.6	15.9	13.1	-3.8	24.3	27

Table 8. Pareto solutions found by the Baron solver and the hybrid algorithm for Instance 1

Row	Baron solver				Hybrid algorithm
	Obj. 1	Obj. 2	Obj. 3 (%)	Obj. 4	Obj. 1	Obj. 2	Obj. 3 (%)	Obj. 4
1	596,320.6	0.2777	0.68	23.3751	223,500	0.58	24.9	22.851
2	441,669.9	0	3.33	22.1017	224,616.6	0.57	24.7	22.798
3	482,948.3	0.2148	1.70	22.2769	236,688.6	0.555	24.2	22.764
4	227,695.7	0.2838	20.44	21.5836	249,431.5	0.54	23.1	22.693
5	269,847.4	0.2532	20.93	21.3166	223,500	0.6	25.7	21.854
6	227,756.6	0.28528	20.44	21.5832	375,100	0.3	25.3	22.903
7	351,791.2	0.40051	9.68	21.9565	223,500	0.6	20.3	22.379
8	268,928.5	0.25386	20.76	21.3141	375,100	0.45	20.6	22.903
9	228,763.6	0.2868	20.45	21.5826	300,388.8	0.494	20.6	21.854
10	228,571.3	0.2871	20.38	21.5841	379,873.8	0	20.6	21.64
11	360,249.7	0.1717	40.14	22.5674	223,500	0.786	20	21.645

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Row	Baron solver				Hybrid algorithm
	Obj. 1	Obj. 2	Obj. 3 (%)	Obj. 4	Obj. 1	Obj. 2	Obj. 3 (%)	Obj. 4
1	596,320.6	0.2777	0.68	23.3751	223,500	0.58	24.9	22.851
2	441,669.9	0	3.33	22.1017	224,616.6	0.57	24.7	22.798
3	482,948.3	0.2148	1.70	22.2769	236,688.6	0.555	24.2	22.764
4	227,695.7	0.2838	20.44	21.5836	249,431.5	0.54	23.1	22.693
5	269,847.4	0.2532	20.93	21.3166	223,500	0.6	25.7	21.854
6	227,756.6	0.28528	20.44	21.5832	375,100	0.3	25.3	22.903
7	351,791.2	0.40051	9.68	21.9565	223,500	0.6	20.3	22.379
8	268,928.5	0.25386	20.76	21.3141	375,100	0.45	20.6	22.903
9	228,763.6	0.2868	20.45	21.5826	300,388.8	0.494	20.6	21.854
10	228,571.3	0.2871	20.38	21.5841	379,873.8	0	20.6	21.64
11	360,249.7	0.1717	40.14	22.5674	223,500	0.786	20	21.645

Table 9. The comparison of the solutions' quality for both the separate optimization and simultaneous optimization

Instance	QM			MID			DM			SM
	Sep.	Sim.	$ \bar{d}_1 $	Sep.	Sim.	$ \bar{d}_2 $	Sep.	Sim.	$ \bar{d}_3 $	Sep.	Sim.	$ \bar{d}_4 $
1	0.600	1	0.400	0.975	0.781	0.194	1.310	1.933	0.623	0.655	1.151	-0.496
2	0.600	1	0.400	1.010	0.586	0.424	1.906	0.739	-1.167	1.454	1.730	-0.276
3	0.500	0.750	0.250	0.994	1.269	-0.275	1.213	1.967	0.754	0.464	0.548	-0.084
4	0.555	0.888	0.333	1.241	1.141	0.100	1.337	1.479	0.142	0.610	0.861	-0.251
5	0.500	1	0.500	1.902	1.432	0.470	1.666	1.479	-0.187	1.037	1.524	-0.487
6	0.428	0.714	0.286	1.342	1.286	0.056	1.555	1.294	-0.261	0.504	0.677	-0.173
7	0.875	0.375	-0.500	1.107	1.287	-0.180	1.576	1.461	-0.115	0.415	0.985	-0.570
8	0.500	1	0.500	0.893	0.624	0.269	1.324	1.636	0.312	0.602	0.854	-0.252
9	0.600	1	0.400	1.365	1.017	0.348	1.521	1.241	-0.280	0.439	0.950	-0.511
10	0.555	0.875	0.320	1.698	1.205	0.493	1.722	1.625	-0.097	0.520	0.991	-0.471
11	0.666	1	0.334	1.031	0.743	0.288	0.883	1.397	0.514	0.999	1.986	-0.987
12	1	1	0	0.863	1.041	-0.178	1.068	0.883	-0.185	0.080	1.278	-1.198
13	0.500	1	0.500	2.631	2.115	0.516	1.536	1.625	0.089	0.268	0.790	-0.522
14	0.666	1	0.334	1.656	1.328	0.328	1.658	1.031	-0.627	0.771	0.790	-0.019
15	0.666	0.666	0	1.246	1.101	0.145	1.521	1.677	0.156	0.950	0.721	0.229
16	0.666	0.500	-0.166	1.462	1.482	-0.020	1.409	1.324	-0.085	0.537	0.746	-0.209
17	0.666	0.666	0	0.877	1.077	-0.200	1.624	1.359	-0.265	0.357	0.472	-0.115
18	0.500	0.777	0.277	0.645	0.639	0.006	1.446	1.365	-0.081	0.698	0.685	0.013
19	0.833	1	0.167	1.791	1.450	0.341	1.552	1.701	0.149	0.578	0.773	-0.195
20	0.666	0.888	0.222	1.308	0.812	0.496	1.291	1.105	-0.186	0.661	0.808	-0.147
Average	0.626	0.854	0.227	1.301	1.120	0.181	1.455	1.415	-0.039	0.629	0.960	-0.336
Gap (%)	36.42			13.91			-2.74			-52.62
Sep: Separate Optimization and Sim: Simultaneous Optimization

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Instance	QM			MID			DM			SM
	Sep.	Sim.	$ \bar{d}_1 $	Sep.	Sim.	$ \bar{d}_2 $	Sep.	Sim.	$ \bar{d}_3 $	Sep.	Sim.	$ \bar{d}_4 $
1	0.600	1	0.400	0.975	0.781	0.194	1.310	1.933	0.623	0.655	1.151	-0.496
2	0.600	1	0.400	1.010	0.586	0.424	1.906	0.739	-1.167	1.454	1.730	-0.276
3	0.500	0.750	0.250	0.994	1.269	-0.275	1.213	1.967	0.754	0.464	0.548	-0.084
4	0.555	0.888	0.333	1.241	1.141	0.100	1.337	1.479	0.142	0.610	0.861	-0.251
5	0.500	1	0.500	1.902	1.432	0.470	1.666	1.479	-0.187	1.037	1.524	-0.487
6	0.428	0.714	0.286	1.342	1.286	0.056	1.555	1.294	-0.261	0.504	0.677	-0.173
7	0.875	0.375	-0.500	1.107	1.287	-0.180	1.576	1.461	-0.115	0.415	0.985	-0.570
8	0.500	1	0.500	0.893	0.624	0.269	1.324	1.636	0.312	0.602	0.854	-0.252
9	0.600	1	0.400	1.365	1.017	0.348	1.521	1.241	-0.280	0.439	0.950	-0.511
10	0.555	0.875	0.320	1.698	1.205	0.493	1.722	1.625	-0.097	0.520	0.991	-0.471
11	0.666	1	0.334	1.031	0.743	0.288	0.883	1.397	0.514	0.999	1.986	-0.987
12	1	1	0	0.863	1.041	-0.178	1.068	0.883	-0.185	0.080	1.278	-1.198
13	0.500	1	0.500	2.631	2.115	0.516	1.536	1.625	0.089	0.268	0.790	-0.522
14	0.666	1	0.334	1.656	1.328	0.328	1.658	1.031	-0.627	0.771	0.790	-0.019
15	0.666	0.666	0	1.246	1.101	0.145	1.521	1.677	0.156	0.950	0.721	0.229
16	0.666	0.500	-0.166	1.462	1.482	-0.020	1.409	1.324	-0.085	0.537	0.746	-0.209
17	0.666	0.666	0	0.877	1.077	-0.200	1.624	1.359	-0.265	0.357	0.472	-0.115
18	0.500	0.777	0.277	0.645	0.639	0.006	1.446	1.365	-0.081	0.698	0.685	0.013
19	0.833	1	0.167	1.791	1.450	0.341	1.552	1.701	0.149	0.578	0.773	-0.195
20	0.666	0.888	0.222	1.308	0.812	0.496	1.291	1.105	-0.186	0.661	0.808	-0.147
Average	0.626	0.854	0.227	1.301	1.120	0.181	1.455	1.415	-0.039	0.629	0.960	-0.336
Gap (%)	36.42			13.91			-2.74			-52.62
Sep: Separate Optimization and Sim: Simultaneous Optimization

Table 10. The comparison of the average unit of the MFT for both the separate optimization and simultaneous optimization

	1	2	3	4	5	6	7	8	9	10
Separate	23.2	42.2	80.1	94.7	124	185	238.3	253.8	295.7	317.6
Simultaneous	22.3	40.6	74.2	90.3	118.9	161.8	211.1	212.7	250.1	265.6
Gap (%)	3.7	3.9	7.4	4.6	4.1	12.5	11.4	16.2	15.4	16.4
	11	12	13	14	15	16	17	18	19	20
Separate	23.1	42.3	73	94.6	126.6	197.4	245.6	266.4	299.9	324.5
Simultaneous	21.8	41	70.6	70.6	118.8	163.6	212.9	223.7	254.7	285.6
Gap (%)	5.5	3.1	3.3	25.4	6.1	17.1	13.3	16	15	11.98

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	1	2	3	4	5	6	7	8	9	10
Separate	23.2	42.2	80.1	94.7	124	185	238.3	253.8	295.7	317.6
Simultaneous	22.3	40.6	74.2	90.3	118.9	161.8	211.1	212.7	250.1	265.6
Gap (%)	3.7	3.9	7.4	4.6	4.1	12.5	11.4	16.2	15.4	16.4
	11	12	13	14	15	16	17	18	19	20
Separate	23.1	42.3	73	94.6	126.6	197.4	245.6	266.4	299.9	324.5
Simultaneous	21.8	41	70.6	70.6	118.8	163.6	212.9	223.7	254.7	285.6
Gap (%)	5.5	3.1	3.3	25.4	6.1	17.1	13.3	16	15	11.98

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American Institute of Mathematical Sciences

An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm

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