Figure 1.
Two possible cases that could happen to determine the start time of a job
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Supervised distance preserving projection using alternating direction method of multipliers
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 |
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
N. Mladenović and P. Hansen,
Variable neighborhood search, Computers & Operations Research, 11 (1997), 1097-1100.
doi: 10.1016/S0305-0548(97)00031-2. |
| [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. |
| [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. |
| [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. |
| [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). |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
N. Mladenović and P. Hansen,
Variable neighborhood search, Computers & Operations Research, 11 (1997), 1097-1100.
doi: 10.1016/S0305-0548(97)00031-2. |
| [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. |
| [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. |
| [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. |
| [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). |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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 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 |
 |
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 |
| 0.7 | 0.8 | 0.9 | 0.7 | 0.8 | 0.9 | 0.7 | 0.8 | 0.9 | |
| 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 |
| 0.7 | 0.8 | 0.9 | 0.7 | 0.8 | 0.9 | 0.7 | 0.8 | 0.9 | |
| 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 |
| 0.7 | 0.7 | 0.8 | |
| 0.2 | 0.3 | 0.3 | |
| Parameter | Size of instances | ||
| Small | Medium | Large-scale | |
| Iteration | 60 | 100 | 150 |
| Initial population | 20 | 30 | 100 |
| 0.7 | 0.7 | 0.8 | |
| 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. | Sep. | Sim. | Sep. | Sim. | Sep. | Sim. | |||||
| 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. | Sep. | Sim. | Sep. | Sim. | Sep. | Sim. | |||||
| 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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