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

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

Corresponding author: Sadoullah Ebrahimnejad*

Received  February 2018 Revised  October 2018 Published  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.

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. 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.  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. HaoM. GenL. 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. KavehV. 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. KhilwaniR. 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. KolischA. 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

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J. LiuD. WangK. 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]

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[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

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N. NekooghadirliR. Tavakkoli-MoghaddamV. 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. SalmaniK. 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. SamarghandiP. 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. ShahrabiM. 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. VelaR. 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. YangS. 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. 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.  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. HaoM. GenL. 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. KavehV. 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. KhilwaniR. 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. KolischA. 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. LiuD. WangK. 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. MohamadiS. 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. NekooghadirliR. Tavakkoli-MoghaddamV. 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. SalmaniK. 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. SamarghandiP. 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. ShahrabiM. 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. VelaR. 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. YangS. 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
Figure 2.  An illustrative example of a solution with 12 departments and 20 jobs
Figure 3.  The candidate locations and departments arrangements for an example with 12 departments
Figure 4.  An illustrative example of the calculation of PUS
Figure 5.  The possible movements, and rotations in the local search for layout
Figure 6.  The flowchart of the second phase of the hybrid algorithm (local search algorithms)
Figure 7.  Illustrative examples of the violation of departments
Figure 8.  Illustration of the solution found for the discrete facility layout of Instance 15
Figure 9.  Illustration of the solution found for the continuous facility layout of Instance 15
Figure 10.  Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 1)
Figure 11.  Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 2)
Figure 12.  The distribution and the interval estimation of the assessment metrics for separate optimization and simultaneous optimization
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
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
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
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)
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
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
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
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
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
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
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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