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A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems
Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization
a. | Center for Food Security and Strategic Studies, Nanjing University of Finance and Economics, 128 Tielubeijie St., Nanjing 210003, China |
b. | Collaborative Innovation Center of Modern Grain Circulation and Safety, Nanjing University of Finance and Economics, 128 Tielubeijie St., Nanjing 210003, China |
c. | School of Mechanical Engineering, Southeast University, Nanjing 211189, China |
Motivated by observing the importance of logistics cost in the cost structure of some products, this paper aims at multi-objective optimization of integrating supply chain network design with the selection of transportation modes (TMs) for a single-product four-echelon supply chain composed of suppliers, production plants, distribution centers (DCs) and customer zones. The key design decisions are the number, capacity and location of plants and DCs, the flow of products through the network, and the selection of TMs for each flow path. A bi-objective mixed integer linear programming model is first formulated. The two incompatible objectives are minimizing the total cost and maximizing the demand fill rate. The model is validated by applying to the case of the design of fresh apple supply chain. Then, a new metaheuristic, called multi-objective modified particle swarm optimization (MMPSO), is presented to find non-dominated solutions. A new modified binary PSO for updating binary variables along with the adaptive mutation is incorporated into the MMPSO. The MMPSO is compared with a multi-objective basic PSO (MBPSO) and the NSGA-Ⅱ against three small cases and six randomly generated medium and large size problems. The comparative results indicate that the MMPSO is better than the NSGA-Ⅱ and the MBPSO with respect to solution quality and computation efficiency for the problem.
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S. H. Amin and G. Zhang,
A multi-objective facility location model for closed-loop supply chain network under uncertain demand and return, Applied Mathematical Modelling, 37 (2013), 4165-4176.
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R. K. Apaiah and E. M. T. Hendrix,
Design of a supply chain network for pea-based novel protein foods, Journal of Food Engineering, 70 (2005), 383-391.
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[4] |
M. Bachlaus, M. Pandey, C. Mahajan, R. Shankar and M. Tiwari,
Designing an integrated multi-echelon agile supply chain network: A hybrid taguchi-particle swarm optimization approach, Journal of Intelligent Manufacturing, 19 (2008), 747-761.
doi: 10.1007/s10845-008-0125-1. |
[5] |
A. Banasik, A. Kanellopoulos, G. Claassen, J. M. Bloemhof-Ruwaard and J. G. van der Vorst,
Closing loops in agricultural supply chains using multi-objective optimization: A case study of an industrial mushroom supply chain, International Journal of Production Economics, 183 (2017), 409-420.
doi: 10.1016/j.ijpe.2016.08.012. |
[6] |
X. Cai, J. Chen, Y. Xiao, X. Xu and G. Yu,
Fresh-product supply chain management with logistics outsourcing, Omega, 41 (2013), 752-765.
doi: 10.1016/j.omega.2012.09.004. |
[7] |
F. T. Chan, A. Jha and M. K. Tiwari,
Bi-objective optimization of three echelon supply chain involving truck selection and loading using NSGA-Ⅱ with heuristics algorithm, Applied Soft Computing, 38 (2016), 978-987.
doi: 10.1016/j.asoc.2015.10.067. |
[8] |
N. Chibeles-Martins, T. Pinto-Varela, A. P. Barbosa-Póvoa and A. Q. Novais,
A multi-objective meta-heuristic approach for the design and planning of green supply chains-MBSA, Expert Systems with Applications, 47 (2016), 71-84.
doi: 10.1016/j.eswa.2015.10.036. |
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C. A. C. Coello, G. T. Pulido and M. S. Lechuga,
Handling multiple objectives with particle swarm optimization, Evolutionary Computation, IEEE Transactions on, 8 (2004), 256-279.
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An integrated model for logistics network design, Annals of Operations Research, 144 (2006), 59-82.
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Machining fixture layout optimisation under dynamic conditions based on evolutionary techniques, International Journal of Production Research, 50 (2012), 4294-4315.
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M. Eskandarpour, P Dejax, J. Miemczyk and O. Péton, Sustainable supply chain network design: An optimization-oriented review, Omega, 54 (2015), 11-32. Google Scholar |
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B. Fahimnia, R. Z. Farahani, R. Marian and L. Luong,
A review and critique on integrated production-distribution planning models and techniques, Journal of Manufacturing Systems, 32 (2013), 1-19.
doi: 10.1016/j.jmsy.2012.07.005. |
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H. Felfel, O. Ayadi and F. Masmoudi,
A decision-making approach for a multi-objective multisite supply network planning problem, International Journal of Computer Integrated Manufacturing, 29 (2016), 754-767.
doi: 10.1080/0951192X.2015.1107916. |
[19] |
K. Govindan, A. Jafarian and V. Nourbakhsh,
Bi-objective integrating sustainable order allocation and sustainable supply chain network strategic design with stochastic demand using a novel robust hybrid multi-objective metaheuristic, Computers & Operations Research, 62 (2015), 112-130.
doi: 10.1016/j.cor.2014.12.014. |
[20] |
G. Guillén, F. Mele, M. Bagajewicz, A. Espuna and L. Puigjaner, Multiobjective supply chain design under uncertainty, Chemical Engineering Science, 60 (2005), 1535-1553. Google Scholar |
[21] |
A. Haddadsisakht and S. M. Ryan,
Closed-loop supply chain network design with multiple transportation modes under stochastic demand and uncertain carbon tax, International Journal of Production Economics, 195 (2018), 118-131.
doi: 10.1016/j.ijpe.2017.09.009. |
[22] |
A. Hafezalkotob, K. Khalili-Damghani and S. S. Ghashami, A Three-Echelon Multi-Objective Multi-Period Multi-Product Supply Chain Network Design Problem: A Goal Programming Approach, Journal of Optimization in Industrial Engineering, 10 (2016), 67-78. Google Scholar |
[23] |
M. Jin, N. A. Granda-Marulanda and I. Down,
The impact of carbon policies on supply chain design and logistics of a major retailer, Journal of Cleaner Production, 85 (2014), 453-461.
doi: 10.1016/j.jclepro.2013.08.042. |
[24] |
F. Jolai, J. Razmi and N. K. M. Rostami,
A fuzzy goal programming and meta heuristic algorithms for solving integrated production: Distribution planning problem, Central European Journal of Operations Research, 19 (2011), 547-569.
doi: 10.1007/s10100-010-0144-9. |
[25] |
R. S. Kadadevaramath, J. C. Chen, B. Latha Shankar and K. Rameshkumar,
Application of particle swarm intelligence algorithms in supply chain network architecture optimization, Expert Systems with Applications, 39 (2012), 10160-10176.
doi: 10.1016/j.eswa.2012.02.116. |
[26] |
M. Kadziński, T. Tervonen, M. K. Tomczyk and R. Dekker, Evaluation of multi-objective optimization approaches for solving green supply chain design problems, Omega, 68 (2017), 168-184. Google Scholar |
[27] |
J. Kennedy and R. C. Eberhart,
A discrete binary version of the particle swarm algorithm, IEEE Press, 5 (1997), 4104-4108.
doi: 10.1109/ICSMC.1997.637339. |
[28] |
J. Kennedy and R. Eberhart,
Particle swarm optimization, Piscataway, NJ: IEEE Service Center, 4 (1995), 1942-1948.
doi: 10.1109/ICNN.1995.488968. |
[29] |
K. Khalili-Damghani, M. Tavana and M. Amirkhan,
A fuzzy bi-objective mixed-integer programming method for solving supply chain network design problems under ambiguous and vague conditions, The International Journal of Advanced Manufacturing Technology, 73 (2014), 1567-1595.
doi: 10.1007/s00170-014-5891-7. |
[30] |
S. Lee, S. Soak, S. Oh, W. Pedrycz and M. Jeon,
Modified binary particle swarm optimization, Progress in Natural Science, 18 (2008), 1161-1166.
doi: 10.1016/j.pnsc.2008.03.018. |
[31] |
M. Mohammadzadeh, A. A. Khamseh and M. Mohammadi,
A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels, Journal of Industrial & Management Optimization, 13 (2017), 1041-1064.
doi: 10.3934/jimo.2016061. |
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L. A. Moncayo-Martínez and D. Z. Zhang, Multi-objective ant colony optimisation: A meta-heuristic approach to supply chain design, International Journal of Production Economics, 131 (2011), 407-420. Google Scholar |
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K. P. Nurjanni, M. S. Carvalho and L. Costa,
Green supply chain design: A mathematical modeling approach based on a multi-objective optimization model, International Journal of Production Economics, 183 (2017), 421-432.
doi: 10.1016/j.ijpe.2016.08.028. |
[34] |
E. Olivares-Benitez, J. L. González-Velarde and R. Z. Ríos-Mercado,
A supply chain design problem with facility location and bi-objective transportation choices, Top, 20 (2012), 729-753.
doi: 10.1007/s11750-010-0162-8. |
[35] |
E. Olivares-Benitez, R. Z. Rios-Mercado and J. L. Gonzalez-Velarde,
A metaheuristic algorithm to solve the selection of transportation channels in supply chain design, International Journal of Production Economics, 145 (2013), 161-172.
doi: 10.1016/j.ijpe.2013.01.017. |
[36] |
K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization method in multiobjective problems, in Proceedings of the 2002 ACM symposium on Applied computing, ACM, (2002), 603–607.
doi: 10.1145/508791.508907. |
[37] |
S. H. R. Pasandideh, S. T. A. Niaki and K. Asadi,
Bi-objective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-Ⅱ and NRGA, Information Sciences, 292 (2015), 57-74.
doi: 10.1016/j.ins.2014.08.068. |
[38] |
M. M. Paydar and M. Saidi-Mehrabad,
Revised multi-choice goal programming for integrated supply chain design and dynamic virtual cell formation with fuzzy parameters, International Journal of Computer Integrated Manufacturing, 28 (2015), 251-265.
doi: 10.1080/0951192X.2013.874596. |
[39] |
M. S. Pishvaee and J. Razmi,
Environmental supply chain network design using multi-objective fuzzy mathematical programming, Applied Mathematical Modelling, 36 (2012), 3433-3446.
doi: 10.1016/j.apm.2011.10.007. |
[40] |
A. Pourrousta, S. Dehbari, R. Tavakkoli-Moghaddam and M. S. Amalnik,
A multi-objective particle swarm optimization for production-distribution planning in supply chain network, Management Science Letters, 2 (2012), 603-614.
doi: 10.5267/j.msl.2011.11.012. |
[41] |
M. Reyes-sierra and C. A. Coello Coello,
Multi-Objective particle swarm optimizers: A survey of the state-of-the-art, International Journal of Computational Intelligence Research, 2 (2006), 287-308.
|
[42] |
H. Sadjady and H. Davoudpour,
Two-echelon, multi-commodity supply chain network design with mode selection, lead-times and inventory costs, Computers & Operations Research, 39 (2012), 1345-1354.
doi: 10.1016/j.cor.2011.08.003. |
[43] |
K. Sarrafha, S. H. A. Rahmati, S. T. A. Niaki and A. Zaretalab,
A bi-objective integrated procurement, production, and distribution problem of a multi-echelon supply chain network design: A new tuned MOEA, Computers & Operations Research, 54 (2015), 35-51.
doi: 10.1016/j.cor.2014.08.010. |
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B. L. Shankar, S. Basavarajappa, J. C. Chen and R. S. Kadadevaramath,
Location and allocation decisions for multi-echelon supply chain network multi-objective evolutionary approach, Expert Systems with Applications, 40 (2013), 551-562.
doi: 10.1016/j.eswa.2012.07.065. |
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C. Shankhar and P. S. S. Prasad,
Cost optimisation of supply chain networks using particle swarm optimisation, International Journal of Business Performance and Supply Chain Modelling, 2 (2010), 112-124.
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show all references
References:
[1] |
F. Altiparmak, M. Gen, L. Lin and T. Paksoy,
A genetic algorithm approach for multi-objective optimization of supply chain networks, Computers & Industrial Engineering, 51 (2006), 196-215.
doi: 10.1016/j.cie.2006.07.011. |
[2] |
S. H. Amin and G. Zhang,
A multi-objective facility location model for closed-loop supply chain network under uncertain demand and return, Applied Mathematical Modelling, 37 (2013), 4165-4176.
doi: 10.1016/j.apm.2012.09.039. |
[3] |
R. K. Apaiah and E. M. T. Hendrix,
Design of a supply chain network for pea-based novel protein foods, Journal of Food Engineering, 70 (2005), 383-391.
doi: 10.1016/j.jfoodeng.2004.02.043. |
[4] |
M. Bachlaus, M. Pandey, C. Mahajan, R. Shankar and M. Tiwari,
Designing an integrated multi-echelon agile supply chain network: A hybrid taguchi-particle swarm optimization approach, Journal of Intelligent Manufacturing, 19 (2008), 747-761.
doi: 10.1007/s10845-008-0125-1. |
[5] |
A. Banasik, A. Kanellopoulos, G. Claassen, J. M. Bloemhof-Ruwaard and J. G. van der Vorst,
Closing loops in agricultural supply chains using multi-objective optimization: A case study of an industrial mushroom supply chain, International Journal of Production Economics, 183 (2017), 409-420.
doi: 10.1016/j.ijpe.2016.08.012. |
[6] |
X. Cai, J. Chen, Y. Xiao, X. Xu and G. Yu,
Fresh-product supply chain management with logistics outsourcing, Omega, 41 (2013), 752-765.
doi: 10.1016/j.omega.2012.09.004. |
[7] |
F. T. Chan, A. Jha and M. K. Tiwari,
Bi-objective optimization of three echelon supply chain involving truck selection and loading using NSGA-Ⅱ with heuristics algorithm, Applied Soft Computing, 38 (2016), 978-987.
doi: 10.1016/j.asoc.2015.10.067. |
[8] |
N. Chibeles-Martins, T. Pinto-Varela, A. P. Barbosa-Póvoa and A. Q. Novais,
A multi-objective meta-heuristic approach for the design and planning of green supply chains-MBSA, Expert Systems with Applications, 47 (2016), 71-84.
doi: 10.1016/j.eswa.2015.10.036. |
[9] |
C. A. C. Coello, G. T. Pulido and M. S. Lechuga,
Handling multiple objectives with particle swarm optimization, Evolutionary Computation, IEEE Transactions on, 8 (2004), 256-279.
doi: 10.1109/TEVC.2004.826067. |
[10] |
C. A. Coello and M. S. Lechuga, MOPSO: A proposal for multiple objective particle swarm optimization Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on, IEEE, (2002), 1051–1056.
doi: 10.1109/CEC.2002.1004388. |
[11] |
J. F. Cordeau, F. Pasin and M. Solomon,
An integrated model for logistics network design, Annals of Operations Research, 144 (2006), 59-82.
doi: 10.1007/s10479-006-0001-3. |
[12] |
K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, 2001. |
[13] |
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan,
A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, Evolutionary Computation, IEEE Transactions on, 6 (2002), 182-197.
doi: 10.1109/4235.996017. |
[14] |
J. Dou, X. Wang and L. Wang,
Machining fixture layout optimisation under dynamic conditions based on evolutionary techniques, International Journal of Production Research, 50 (2012), 4294-4315.
doi: 10.1080/00207543.2011.618470. |
[15] |
J. J. Durillo, J. García-Nieto, A. J. Nebro, C. A. C. Coello, F. Luna and E. Alba, Multiobjective particle swarm optimizers: An experimental comparison, Evolutionary MultiCriterion Optimization, Springer, (2009), 495–509.
doi: 10.1007/978-3-642-01020-0_39. |
[16] |
M. Eskandarpour, P Dejax, J. Miemczyk and O. Péton, Sustainable supply chain network design: An optimization-oriented review, Omega, 54 (2015), 11-32. Google Scholar |
[17] |
B. Fahimnia, R. Z. Farahani, R. Marian and L. Luong,
A review and critique on integrated production-distribution planning models and techniques, Journal of Manufacturing Systems, 32 (2013), 1-19.
doi: 10.1016/j.jmsy.2012.07.005. |
[18] |
H. Felfel, O. Ayadi and F. Masmoudi,
A decision-making approach for a multi-objective multisite supply network planning problem, International Journal of Computer Integrated Manufacturing, 29 (2016), 754-767.
doi: 10.1080/0951192X.2015.1107916. |
[19] |
K. Govindan, A. Jafarian and V. Nourbakhsh,
Bi-objective integrating sustainable order allocation and sustainable supply chain network strategic design with stochastic demand using a novel robust hybrid multi-objective metaheuristic, Computers & Operations Research, 62 (2015), 112-130.
doi: 10.1016/j.cor.2014.12.014. |
[20] |
G. Guillén, F. Mele, M. Bagajewicz, A. Espuna and L. Puigjaner, Multiobjective supply chain design under uncertainty, Chemical Engineering Science, 60 (2005), 1535-1553. Google Scholar |
[21] |
A. Haddadsisakht and S. M. Ryan,
Closed-loop supply chain network design with multiple transportation modes under stochastic demand and uncertain carbon tax, International Journal of Production Economics, 195 (2018), 118-131.
doi: 10.1016/j.ijpe.2017.09.009. |
[22] |
A. Hafezalkotob, K. Khalili-Damghani and S. S. Ghashami, A Three-Echelon Multi-Objective Multi-Period Multi-Product Supply Chain Network Design Problem: A Goal Programming Approach, Journal of Optimization in Industrial Engineering, 10 (2016), 67-78. Google Scholar |
[23] |
M. Jin, N. A. Granda-Marulanda and I. Down,
The impact of carbon policies on supply chain design and logistics of a major retailer, Journal of Cleaner Production, 85 (2014), 453-461.
doi: 10.1016/j.jclepro.2013.08.042. |
[24] |
F. Jolai, J. Razmi and N. K. M. Rostami,
A fuzzy goal programming and meta heuristic algorithms for solving integrated production: Distribution planning problem, Central European Journal of Operations Research, 19 (2011), 547-569.
doi: 10.1007/s10100-010-0144-9. |
[25] |
R. S. Kadadevaramath, J. C. Chen, B. Latha Shankar and K. Rameshkumar,
Application of particle swarm intelligence algorithms in supply chain network architecture optimization, Expert Systems with Applications, 39 (2012), 10160-10176.
doi: 10.1016/j.eswa.2012.02.116. |
[26] |
M. Kadziński, T. Tervonen, M. K. Tomczyk and R. Dekker, Evaluation of multi-objective optimization approaches for solving green supply chain design problems, Omega, 68 (2017), 168-184. Google Scholar |
[27] |
J. Kennedy and R. C. Eberhart,
A discrete binary version of the particle swarm algorithm, IEEE Press, 5 (1997), 4104-4108.
doi: 10.1109/ICSMC.1997.637339. |
[28] |
J. Kennedy and R. Eberhart,
Particle swarm optimization, Piscataway, NJ: IEEE Service Center, 4 (1995), 1942-1948.
doi: 10.1109/ICNN.1995.488968. |
[29] |
K. Khalili-Damghani, M. Tavana and M. Amirkhan,
A fuzzy bi-objective mixed-integer programming method for solving supply chain network design problems under ambiguous and vague conditions, The International Journal of Advanced Manufacturing Technology, 73 (2014), 1567-1595.
doi: 10.1007/s00170-014-5891-7. |
[30] |
S. Lee, S. Soak, S. Oh, W. Pedrycz and M. Jeon,
Modified binary particle swarm optimization, Progress in Natural Science, 18 (2008), 1161-1166.
doi: 10.1016/j.pnsc.2008.03.018. |
[31] |
M. Mohammadzadeh, A. A. Khamseh and M. Mohammadi,
A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels, Journal of Industrial & Management Optimization, 13 (2017), 1041-1064.
doi: 10.3934/jimo.2016061. |
[32] |
L. A. Moncayo-Martínez and D. Z. Zhang, Multi-objective ant colony optimisation: A meta-heuristic approach to supply chain design, International Journal of Production Economics, 131 (2011), 407-420. Google Scholar |
[33] |
K. P. Nurjanni, M. S. Carvalho and L. Costa,
Green supply chain design: A mathematical modeling approach based on a multi-objective optimization model, International Journal of Production Economics, 183 (2017), 421-432.
doi: 10.1016/j.ijpe.2016.08.028. |
[34] |
E. Olivares-Benitez, J. L. González-Velarde and R. Z. Ríos-Mercado,
A supply chain design problem with facility location and bi-objective transportation choices, Top, 20 (2012), 729-753.
doi: 10.1007/s11750-010-0162-8. |
[35] |
E. Olivares-Benitez, R. Z. Rios-Mercado and J. L. Gonzalez-Velarde,
A metaheuristic algorithm to solve the selection of transportation channels in supply chain design, International Journal of Production Economics, 145 (2013), 161-172.
doi: 10.1016/j.ijpe.2013.01.017. |
[36] |
K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization method in multiobjective problems, in Proceedings of the 2002 ACM symposium on Applied computing, ACM, (2002), 603–607.
doi: 10.1145/508791.508907. |
[37] |
S. H. R. Pasandideh, S. T. A. Niaki and K. Asadi,
Bi-objective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-Ⅱ and NRGA, Information Sciences, 292 (2015), 57-74.
doi: 10.1016/j.ins.2014.08.068. |
[38] |
M. M. Paydar and M. Saidi-Mehrabad,
Revised multi-choice goal programming for integrated supply chain design and dynamic virtual cell formation with fuzzy parameters, International Journal of Computer Integrated Manufacturing, 28 (2015), 251-265.
doi: 10.1080/0951192X.2013.874596. |
[39] |
M. S. Pishvaee and J. Razmi,
Environmental supply chain network design using multi-objective fuzzy mathematical programming, Applied Mathematical Modelling, 36 (2012), 3433-3446.
doi: 10.1016/j.apm.2011.10.007. |
[40] |
A. Pourrousta, S. Dehbari, R. Tavakkoli-Moghaddam and M. S. Amalnik,
A multi-objective particle swarm optimization for production-distribution planning in supply chain network, Management Science Letters, 2 (2012), 603-614.
doi: 10.5267/j.msl.2011.11.012. |
[41] |
M. Reyes-sierra and C. A. Coello Coello,
Multi-Objective particle swarm optimizers: A survey of the state-of-the-art, International Journal of Computational Intelligence Research, 2 (2006), 287-308.
|
[42] |
H. Sadjady and H. Davoudpour,
Two-echelon, multi-commodity supply chain network design with mode selection, lead-times and inventory costs, Computers & Operations Research, 39 (2012), 1345-1354.
doi: 10.1016/j.cor.2011.08.003. |
[43] |
K. Sarrafha, S. H. A. Rahmati, S. T. A. Niaki and A. Zaretalab,
A bi-objective integrated procurement, production, and distribution problem of a multi-echelon supply chain network design: A new tuned MOEA, Computers & Operations Research, 54 (2015), 35-51.
doi: 10.1016/j.cor.2014.08.010. |
[44] |
B. L. Shankar, S. Basavarajappa, J. C. Chen and R. S. Kadadevaramath,
Location and allocation decisions for multi-echelon supply chain network multi-objective evolutionary approach, Expert Systems with Applications, 40 (2013), 551-562.
doi: 10.1016/j.eswa.2012.07.065. |
[45] |
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# | Plant locations | DC locations | |||||||||||
YA | TC | BO | XY | WN | XA | BJ | SH | SZ | WH | ZZ | CD | XA | |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
2 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
3 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
4 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
5 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Demand | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA | ||||
1 | 24000 | 32000 | 16000 | 32000 | 40000 | 24000 | 32000 | 24000 | 40000 | ||||
2 | 24000 | 32000 | 16287.6 | 33855.8 | 41188 | 27999.6 | 36140.1 | 24000 | 40000 | ||||
3 | 24000 | 32000 | 19765.1 | 35230.7 | 48442 | 25584.1 | 33739.3 | 27232.1 | 50000 | ||||
4 | 30000 | 32687.3 | 20000 | 40000 | 46619.4 | 30000 | 39736.1 | 24000 | 50000 | ||||
5 | 30000 | 40000 | 20000 | 40000 | 50000 | 30000 | 40000 | 30000 | 50000 | ||||
TM | Supplier-to-Plant | Plant-to-DC | DC-to-CZ | ||||||||||
1 | truck | rail | rail+truck | ||||||||||
2 | truck | rail | rail+truck | ||||||||||
3 | truck | rail+truck | rail+truck | ||||||||||
4 | truck | rail+truck | rail+truck | ||||||||||
5 | truck | rail+truck | rail+truck | ||||||||||
# | 1 | 2 | 3 | 4 | 5 | ||||||||
Cost/1E8 | 4.7783 | 5.0033 | 5.3648 | 5.6808 | 5.9819 | ||||||||
DFR | 0.8 | 0.835 | 0.897 | 0.949 | 1.0 |
# | Plant locations | DC locations | |||||||||||
YA | TC | BO | XY | WN | XA | BJ | SH | SZ | WH | ZZ | CD | XA | |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
2 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
3 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
4 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
5 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Demand | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA | ||||
1 | 24000 | 32000 | 16000 | 32000 | 40000 | 24000 | 32000 | 24000 | 40000 | ||||
2 | 24000 | 32000 | 16287.6 | 33855.8 | 41188 | 27999.6 | 36140.1 | 24000 | 40000 | ||||
3 | 24000 | 32000 | 19765.1 | 35230.7 | 48442 | 25584.1 | 33739.3 | 27232.1 | 50000 | ||||
4 | 30000 | 32687.3 | 20000 | 40000 | 46619.4 | 30000 | 39736.1 | 24000 | 50000 | ||||
5 | 30000 | 40000 | 20000 | 40000 | 50000 | 30000 | 40000 | 30000 | 50000 | ||||
TM | Supplier-to-Plant | Plant-to-DC | DC-to-CZ | ||||||||||
1 | truck | rail | rail+truck | ||||||||||
2 | truck | rail | rail+truck | ||||||||||
3 | truck | rail+truck | rail+truck | ||||||||||
4 | truck | rail+truck | rail+truck | ||||||||||
5 | truck | rail+truck | rail+truck | ||||||||||
# | 1 | 2 | 3 | 4 | 5 | ||||||||
Cost/1E8 | 4.7783 | 5.0033 | 5.3648 | 5.6808 | 5.9819 | ||||||||
DFR | 0.8 | 0.835 | 0.897 | 0.949 | 1.0 |
CZs | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
Apples | 30 | 40 | 20 | 40 | 50 | 30 | 40 | 30 | 50 |
CZs | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
Apples | 30 | 40 | 20 | 40 | 50 | 30 | 40 | 30 | 50 |
Locations | YA | TC | BO | XY | WN |
Capacity (K tons) | 120 | 60 | 60 | 160 | 100 |
cost (RMB/ton) | 610 | 620 | 630 | 620 | 630 |
Locations | YA | TC | BO | XY | WN |
Capacity (K tons) | 120 | 60 | 60 | 160 | 100 |
cost (RMB/ton) | 610 | 620 | 630 | 620 | 630 |
Sites | YA | TC | BO | XY | WN | XA |
Cost | 500 | 520 | 540 | 530 | 530 | 520 |
Sites | YA | TC | BO | XY | WN | XA |
Cost | 500 | 520 | 540 | 530 | 530 | 520 |
DCs | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
Cost | 420 | 430 | 440 | 410 | 400 | 400 | 400 | 400 | 400 |
DCs | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
Cost | 420 | 430 | 440 | 410 | 400 | 400 | 400 | 400 | 400 |
cost | YA | TC | BO | XY | WN | XA |
YA | 0 | 103.2 | 212.4 | 156 | 139.8 | 157.2 |
TC | 103.2 | 0 | 135 | 53.4 | 48 | 53.4 |
BO | 212.4 | 135 | 0 | 99.6 | 153 | 120 |
XY | 156 | 53.4 | 99.6 | 0 | 53.4 | 21 |
WN | 139.8 | 48 | 153 | 53.4 | 0 | 35.4 |
cost | YA | TC | BO | XY | WN | XA |
YA | 0 | 103.2 | 212.4 | 156 | 139.8 | 157.2 |
TC | 103.2 | 0 | 135 | 53.4 | 48 | 53.4 |
BO | 212.4 | 135 | 0 | 99.6 | 153 | 120 |
XY | 156 | 53.4 | 99.6 | 0 | 53.4 | 21 |
WN | 139.8 | 48 | 153 | 53.4 | 0 | 35.4 |
cost | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
YA | 127.6 | 150.8 | 199.6 | 116.1 | 143.3 | 75.9 | 100.8 | 93.9 | 37.6 |
516 | 877.2 | 984.6 | 517.8 | 607.8 | 298.2 | 533.4 | 508.8 | 157.2 | |
TC | 110.2 | 133.4 | 182.2 | 98.7 | 125.9 | 58.5 | 83.4 | 78.5 | 20.2 |
588 | 866.4 | 899.4 | 467.4 | 529.2 | 300.6 | 444 | 406.2 | 53.4 | |
BO | 114.2 | 137.4 | 186.2 | 102.7 | 129.9 | 62.5 | 61.4 | 82.5 | 24.2 |
720.6 | 976.8 | 912.6 | 543 | 566.4 | 427.2 | 322.2 | 324 | 120 | |
XY | 102.9 | 126.1 | 174.9 | 91.5 | 118.6 | 51.3 | 72.6 | 67.8 | 12.9 |
636.6 | 879.6 | 866.4 | 454.8 | 503.4 | 327.6 | 393.6 | 352.2 | 21 | |
WN | 97 | 120.2 | 187.7 | 85.5 | 112.7 | 45.3 | 78.6 | 73.7 | 15.4 |
589.8 | 829.2 | 852 | 416.4 | 481.2 | 273.6 | 440.4 | 383.4 | 35.4 | |
XA | 101.2 | 124.4 | 173.2 | 90 | 116.9 | 49.5 | 74.4 | 69.5 | 11.2 |
624 | 858.6 | 852.6 | 435 | 486.6 | 308.4 | 405.6 | 355.2 | 0 | |
BJ | 0 | 120.3 | 189.1 | 102.7 | 130.7 | 62.9 | 164.4 | 167 | 101.2 |
0 | 670.2 | 1167 | 645 | 816 | 391.8 | 1030.2 | 958.2 | 624 | |
SH | 120.3 | 0 | 137.6 | 72.8 | 99.6 | 86.1 | 172.5 | 173.8 | 124.4 |
670.2 | 0 | 757.2 | 481.8 | 600.6 | 578.4 | 1159.8 | 1005.6 | 858.6 | |
SZ | 189.1 | 137.6 | 0 | 103.9 | 75.3 | 153.5 | 187.9 | 164.3 | 173.2 |
1167 | 757.2 | 0 | 528 | 381 | 816 | 858.6 | 689.4 | 852.6 | |
WH | 102.7 | 72.8 | 103.9 | 0 | 38.3 | 51.4 | 113.1 | 102.7 | 90 |
645 | 481.8 | 528 | 0 | 176.4 | 291 | 679.8 | 523.2 | 435 | |
CS | 130.7 | 99.6 | 75.3 | 38.3 | 0 | 78.6 | 112.9 | 93.3 | 116.9 |
816 | 600.6 | 381 | 176.4 | 0 | 442.8 | 616.2 | 442.8 | 486.6 | |
ZZ | 62.9 | 86.1 | 153.5 | 51.4 | 78.6 | 0 | 112.7 | 115.5 | 49.5 |
391.8 | 578.4 | 816 | 291 | 442.8 | 0 | 690 | 586.8 | 308.4 | |
CD | 164.4 | 172.5 | 187.9 | 113.1 | 112.9 | 112.7 | 0 | 34.7 | 74.4 |
1030.2 | 1159.8 | 858.6 | 679.8 | 616.2 | 690 | 0 | 175.8 | 405.6 | |
CQ | 167 | 173.8 | 164.3 | 102.7 | 93.3 | 115.5 | 34.7 | 0 | 69.5 |
958.2 | 1005.6 | 689.4 | 523.2 | 442.8 | 586.8 | 175.8 | 0 | 355.2 | |
Note: the italic numbers denote unit cost of rail, and non-italic numbers denote unit cost of truck. |
cost | BJ | SH | SZ | WH | CS | ZZ | CD | CQ | XA |
YA | 127.6 | 150.8 | 199.6 | 116.1 | 143.3 | 75.9 | 100.8 | 93.9 | 37.6 |
516 | 877.2 | 984.6 | 517.8 | 607.8 | 298.2 | 533.4 | 508.8 | 157.2 | |
TC | 110.2 | 133.4 | 182.2 | 98.7 | 125.9 | 58.5 | 83.4 | 78.5 | 20.2 |
588 | 866.4 | 899.4 | 467.4 | 529.2 | 300.6 | 444 | 406.2 | 53.4 | |
BO | 114.2 | 137.4 | 186.2 | 102.7 | 129.9 | 62.5 | 61.4 | 82.5 | 24.2 |
720.6 | 976.8 | 912.6 | 543 | 566.4 | 427.2 | 322.2 | 324 | 120 | |
XY | 102.9 | 126.1 | 174.9 | 91.5 | 118.6 | 51.3 | 72.6 | 67.8 | 12.9 |
636.6 | 879.6 | 866.4 | 454.8 | 503.4 | 327.6 | 393.6 | 352.2 | 21 | |
WN | 97 | 120.2 | 187.7 | 85.5 | 112.7 | 45.3 | 78.6 | 73.7 | 15.4 |
589.8 | 829.2 | 852 | 416.4 | 481.2 | 273.6 | 440.4 | 383.4 | 35.4 | |
XA | 101.2 | 124.4 | 173.2 | 90 | 116.9 | 49.5 | 74.4 | 69.5 | 11.2 |
624 | 858.6 | 852.6 | 435 | 486.6 | 308.4 | 405.6 | 355.2 | 0 | |
BJ | 0 | 120.3 | 189.1 | 102.7 | 130.7 | 62.9 | 164.4 | 167 | 101.2 |
0 | 670.2 | 1167 | 645 | 816 | 391.8 | 1030.2 | 958.2 | 624 | |
SH | 120.3 | 0 | 137.6 | 72.8 | 99.6 | 86.1 | 172.5 | 173.8 | 124.4 |
670.2 | 0 | 757.2 | 481.8 | 600.6 | 578.4 | 1159.8 | 1005.6 | 858.6 | |
SZ | 189.1 | 137.6 | 0 | 103.9 | 75.3 | 153.5 | 187.9 | 164.3 | 173.2 |
1167 | 757.2 | 0 | 528 | 381 | 816 | 858.6 | 689.4 | 852.6 | |
WH | 102.7 | 72.8 | 103.9 | 0 | 38.3 | 51.4 | 113.1 | 102.7 | 90 |
645 | 481.8 | 528 | 0 | 176.4 | 291 | 679.8 | 523.2 | 435 | |
CS | 130.7 | 99.6 | 75.3 | 38.3 | 0 | 78.6 | 112.9 | 93.3 | 116.9 |
816 | 600.6 | 381 | 176.4 | 0 | 442.8 | 616.2 | 442.8 | 486.6 | |
ZZ | 62.9 | 86.1 | 153.5 | 51.4 | 78.6 | 0 | 112.7 | 115.5 | 49.5 |
391.8 | 578.4 | 816 | 291 | 442.8 | 0 | 690 | 586.8 | 308.4 | |
CD | 164.4 | 172.5 | 187.9 | 113.1 | 112.9 | 112.7 | 0 | 34.7 | 74.4 |
1030.2 | 1159.8 | 858.6 | 679.8 | 616.2 | 690 | 0 | 175.8 | 405.6 | |
CQ | 167 | 173.8 | 164.3 | 102.7 | 93.3 | 115.5 | 34.7 | 0 | 69.5 |
958.2 | 1005.6 | 689.4 | 523.2 | 442.8 | 586.8 | 175.8 | 0 | 355.2 | |
Note: the italic numbers denote unit cost of rail, and non-italic numbers denote unit cost of truck. |
Pro. | Item | ER | CT/seconds | |||||||
MMPSO | MBPSO | NSGA-Ⅱ | MMPSO | MBPSO | NSGA-Ⅱ | MMPSO | MBPSO | NSGA-Ⅱ | ||
Pro 4 |
avg. | 0.367 | 0.96 | 0.52 | 767.1 | 784.9 | 798.3 | 0.303 | 0.455 | 0.436 |
best | 0.3 | 0.933 | 0.13 | 737.6 | 762.3 | 773.1 | 0.199 | 0.394 | 0.373 | |
std. | 0.0782 | 0.0279 | 0.288 | 35.8 | 20 | 29.3 | 0.049 | 0.047 | 0.051 | |
Pro 5 |
avg. | 0.513 | 0.9 | 0.547 | 1147.9 | 1193.8 | 1174.6 | 0.318 | 0.714 | 0.453 |
best | 0.433 | 0.8 | 0.433 | 1091.6 | 1138.7 | 1131 | 0.266 | 0.695 | 0.408 | |
std. | 0.0506 | 0.085 | 0.0989 | 36.7 | 41.2 | 41.6 | 0.039 | 0.014 | 0.043 | |
Pro 6 |
avg. | 0.487 | 0.953 | 0.52 | 1487.6 | 1472.1 | 1491.9 | 0.315 | 0.502 | 0.461 |
best | 0.433 | 0.9 | 0.3 | 1424.8 | 1421.8 | 1404.4 | 0.232 | 0.358 | 0.412 | |
std. | 0.0581 | 0.04 | 0.1609 | 46.4 | 38.69 | 100.8 | 0.062 | 0.084 | 0.041 | |
Pro 7 |
avg. | 0.488 | 0.958 | 0.519 | 1511.4 | 1529.3 | 1538.9 | 0.304 | 0.436 | 0.423 |
best | 0.4 | 0.85 | 0.35 | 1484.8 | 1494.3 | 1526.6 | 0.239 | 0.356 | 0.378 | |
std. | 0.0598 | 0.054 | 0.165 | 16.29 | 38.03 | 12.52 | 0.059 | 0.042 | 0.037 | |
Pro 8 |
avg. | 0.48 | 0.946 | 0.48 | 1947.2 | 1953.6 | 1951.6 | 0.346 | 0.506 | 0.435 |
best | 0.375 | 0.9 | 0.4 | 1933.2 | 1932.8 | 1944.6 | 0.292 | 0.421 | 0.302 | |
std. | 0.087 | 0.025 | 0.043 | 13.7 | 14.5 | 6.43 | 0.041 | 0.046 | 0.057 | |
Pro 9 |
avg. | 0.472 | 0.958 | 0.544 | 2351.2 | 2383 | 2355.8 | 0.323 | 0.509 | 0.413 |
best | 0.35 | 0.95 | 0.325 | 2343 | 2316.3 | 2348.1 | 0.239 | 0.432 | 0.303 | |
std. | 0.078 | 0.013 | 0.188 | 7.62 | 9.21 | 6.74 | 0.079 | 0.053 | 0.069 |
Pro. | Item | ER | CT/seconds | |||||||
MMPSO | MBPSO | NSGA-Ⅱ | MMPSO | MBPSO | NSGA-Ⅱ | MMPSO | MBPSO | NSGA-Ⅱ | ||
Pro 4 |
avg. | 0.367 | 0.96 | 0.52 | 767.1 | 784.9 | 798.3 | 0.303 | 0.455 | 0.436 |
best | 0.3 | 0.933 | 0.13 | 737.6 | 762.3 | 773.1 | 0.199 | 0.394 | 0.373 | |
std. | 0.0782 | 0.0279 | 0.288 | 35.8 | 20 | 29.3 | 0.049 | 0.047 | 0.051 | |
Pro 5 |
avg. | 0.513 | 0.9 | 0.547 | 1147.9 | 1193.8 | 1174.6 | 0.318 | 0.714 | 0.453 |
best | 0.433 | 0.8 | 0.433 | 1091.6 | 1138.7 | 1131 | 0.266 | 0.695 | 0.408 | |
std. | 0.0506 | 0.085 | 0.0989 | 36.7 | 41.2 | 41.6 | 0.039 | 0.014 | 0.043 | |
Pro 6 |
avg. | 0.487 | 0.953 | 0.52 | 1487.6 | 1472.1 | 1491.9 | 0.315 | 0.502 | 0.461 |
best | 0.433 | 0.9 | 0.3 | 1424.8 | 1421.8 | 1404.4 | 0.232 | 0.358 | 0.412 | |
std. | 0.0581 | 0.04 | 0.1609 | 46.4 | 38.69 | 100.8 | 0.062 | 0.084 | 0.041 | |
Pro 7 |
avg. | 0.488 | 0.958 | 0.519 | 1511.4 | 1529.3 | 1538.9 | 0.304 | 0.436 | 0.423 |
best | 0.4 | 0.85 | 0.35 | 1484.8 | 1494.3 | 1526.6 | 0.239 | 0.356 | 0.378 | |
std. | 0.0598 | 0.054 | 0.165 | 16.29 | 38.03 | 12.52 | 0.059 | 0.042 | 0.037 | |
Pro 8 |
avg. | 0.48 | 0.946 | 0.48 | 1947.2 | 1953.6 | 1951.6 | 0.346 | 0.506 | 0.435 |
best | 0.375 | 0.9 | 0.4 | 1933.2 | 1932.8 | 1944.6 | 0.292 | 0.421 | 0.302 | |
std. | 0.087 | 0.025 | 0.043 | 13.7 | 14.5 | 6.43 | 0.041 | 0.046 | 0.057 | |
Pro 9 |
avg. | 0.472 | 0.958 | 0.544 | 2351.2 | 2383 | 2355.8 | 0.323 | 0.509 | 0.413 |
best | 0.35 | 0.95 | 0.325 | 2343 | 2316.3 | 2348.1 | 0.239 | 0.432 | 0.303 | |
std. | 0.078 | 0.013 | 0.188 | 7.62 | 9.21 | 6.74 | 0.079 | 0.053 | 0.069 |
Problem | # | I | J | K | S | P | M |
Medium size | 4 | 8 | 10 | 12 | 19 | 3 | 3 |
5 | 10 | 12 | 14 | 23 | 3 | 3 | |
6 | 12 | 14 | 15 | 25 | 3 | 3 | |
Large size | 7 | 14 | 15 | 16 | 29 | 5 | 4 |
8 | 15 | 18 | 18 | 33 | 5 | 4 | |
9 | 17 | 19 | 20 | 35 | 5 | 4 |
Problem | # | I | J | K | S | P | M |
Medium size | 4 | 8 | 10 | 12 | 19 | 3 | 3 |
5 | 10 | 12 | 14 | 23 | 3 | 3 | |
6 | 12 | 14 | 15 | 25 | 3 | 3 | |
Large size | 7 | 14 | 15 | 16 | 29 | 5 | 4 |
8 | 15 | 18 | 18 | 33 | 5 | 4 | |
9 | 17 | 19 | 20 | 35 | 5 | 4 |
Plant locations | DC locations | |||||||||
YA | TC | BO | XY | WN | BJ | SH | SZ | ZZ | WH | XA |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 |
TM Selection | ||||||||||
Optional TMs | Supplier-to-Plant | Plant-to-DC | DC-to-CZ | |||||||
Truck only | Truck | Truck | Truck | |||||||
Truck and Rail | Truck | Rail | Rail | |||||||
Note: the first row with italic numbers is binary decision without TM selection, and the second row is with TM selection |
Plant locations | DC locations | |||||||||
YA | TC | BO | XY | WN | BJ | SH | SZ | ZZ | WH | XA |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 |
TM Selection | ||||||||||
Optional TMs | Supplier-to-Plant | Plant-to-DC | DC-to-CZ | |||||||
Truck only | Truck | Truck | Truck | |||||||
Truck and Rail | Truck | Rail | Rail | |||||||
Note: the first row with italic numbers is binary decision without TM selection, and the second row is with TM selection |
Case | Metric | Algorithm | Run 1 | Run 2 | Run 3 | Run 4 | Run 5 | Average | Best |
Case1 | ER | MMPSO | 15/20 | 12/20 | 8/20 | 3/20 | 7/20 | ||
MBPSO | 19/20 | 11/20 | 12/20 | 14/20 | 16/20 | 72% | 55% | ||
NSGA-Ⅱ | 11/20 | 8/20 | 10/20 | 12/20 | 10/20 | 51% | 40% | ||
CT/seconds | MMPSO | 46.550 | 49.901 | 45.615 | 41.948 | 48.298 | 46.462 | 41.948 | |
MBPSO | 53.029 | 54.990 | 51.532 | 54.416 | 52.358 | 53.265 | 51.532 | ||
NSGA-Ⅱ | 56.441 | 53.788 | 52.416 | 52.791 | 54.148 | 53.917 | 52.416 | ||
MMPSO | 0.478 | 0.467 | 0.313 | 0.322 | 0.462 | 0.408 | 0.313 | ||
MBPSO | 0.290 | 0.326 | 0.357 | 0.339 | 0.315 | 0.326 | 0.290 | ||
NSGA-Ⅱ | 0.580 | 0.470 | 0.626 | 0.426 | 0.645 | 0.549 | 0.426 | ||
Case2 | ER | MMPSO | 11/20 | 8/20 | 9/20 | 8/20 | 11/20 | 47% | 40% |
MBPSO | 20/20 | 15/20 | 18/20 | 16/20 | 16/20 | 85% | 75% | ||
NSGA-Ⅱ | 11/20 | 12/20 | 12/20 | 10/20 | 9/20 | 54% | 50% | ||
CT/seconds | MMPSO | 54.413 | 52.953 | 54.179 | 59.585 | 59.405 | 56.107 | 52.953 | |
MBPSO | 50.104 | 53.325 | 55.099 | 60.025 | 54.380 | 54.587 | 50.104 | ||
NSGA-Ⅱ | 56.471 | 58.846 | 60.847 | 55.968 | 57.705 | 57.967 | 55.968 | ||
MMPSO | 0.283 | 0.369 | 0.326 | 0.275 | 0.341 | 0.319 | 0.275 | ||
MBPSO | 0.329 | 0.353 | 0.269 | 0.391 | 0.357 | 0.339 | 0.269 | ||
NSGA-Ⅱ | 0.631 | 0.578 | 0.455 | 0.517 | 0.503 | 0.537 | 0.455 | ||
Case3 | ER | MMPSO | 8/20 | 9/20 | 8/20 | 14/20 | 8/20 | 47% | 40% |
MBPSO | 17/20 | 14/20 | 17/20 | 13/20 | 16/20 | 77% | 65% | ||
NSGA-Ⅱ | 7/20 | 10/20 | 12/20 | 12/20 | 10/20 | 51% | 35% | ||
CT/seconds | MMPSO | 43.607 | 43.898 | 37.106 | 39.952 | 42.460 | 41.405 | 37.106 | |
MBPSO | 43.612 | 42.369 | 43.306 | 42.728 | 48.301 | 44.063 | 42.369 | ||
NSGA-Ⅱ | 45.038 | 48.282 | 44.382 | 45.383 | 46.448 | 45.907 | 44.382 | ||
MMPSO | 0.365 | 0.257 | 0.373 | 0.338 | 0.286 | 0.324 | 0.257 | ||
MBPSO | 0.412 | 0.379 | 0.408 | 0.292 | 0.238 | 0.346 | 0.238 | ||
NSGA-Ⅱ | 0.536 | 0.544 | 0.562 | 0.498 | 0.571 | 0.542 | 0.498 |
Case | Metric | Algorithm | Run 1 | Run 2 | Run 3 | Run 4 | Run 5 | Average | Best |
Case1 | ER | MMPSO | 15/20 | 12/20 | 8/20 | 3/20 | 7/20 | ||
MBPSO | 19/20 | 11/20 | 12/20 | 14/20 | 16/20 | 72% | 55% | ||
NSGA-Ⅱ | 11/20 | 8/20 | 10/20 | 12/20 | 10/20 | 51% | 40% | ||
CT/seconds | MMPSO | 46.550 | 49.901 | 45.615 | 41.948 | 48.298 | 46.462 | 41.948 | |
MBPSO | 53.029 | 54.990 | 51.532 | 54.416 | 52.358 | 53.265 | 51.532 | ||
NSGA-Ⅱ | 56.441 | 53.788 | 52.416 | 52.791 | 54.148 | 53.917 | 52.416 | ||
MMPSO | 0.478 | 0.467 | 0.313 | 0.322 | 0.462 | 0.408 | 0.313 | ||
MBPSO | 0.290 | 0.326 | 0.357 | 0.339 | 0.315 | 0.326 | 0.290 | ||
NSGA-Ⅱ | 0.580 | 0.470 | 0.626 | 0.426 | 0.645 | 0.549 | 0.426 | ||
Case2 | ER | MMPSO | 11/20 | 8/20 | 9/20 | 8/20 | 11/20 | 47% | 40% |
MBPSO | 20/20 | 15/20 | 18/20 | 16/20 | 16/20 | 85% | 75% | ||
NSGA-Ⅱ | 11/20 | 12/20 | 12/20 | 10/20 | 9/20 | 54% | 50% | ||
CT/seconds | MMPSO | 54.413 | 52.953 | 54.179 | 59.585 | 59.405 | 56.107 | 52.953 | |
MBPSO | 50.104 | 53.325 | 55.099 | 60.025 | 54.380 | 54.587 | 50.104 | ||
NSGA-Ⅱ | 56.471 | 58.846 | 60.847 | 55.968 | 57.705 | 57.967 | 55.968 | ||
MMPSO | 0.283 | 0.369 | 0.326 | 0.275 | 0.341 | 0.319 | 0.275 | ||
MBPSO | 0.329 | 0.353 | 0.269 | 0.391 | 0.357 | 0.339 | 0.269 | ||
NSGA-Ⅱ | 0.631 | 0.578 | 0.455 | 0.517 | 0.503 | 0.537 | 0.455 | ||
Case3 | ER | MMPSO | 8/20 | 9/20 | 8/20 | 14/20 | 8/20 | 47% | 40% |
MBPSO | 17/20 | 14/20 | 17/20 | 13/20 | 16/20 | 77% | 65% | ||
NSGA-Ⅱ | 7/20 | 10/20 | 12/20 | 12/20 | 10/20 | 51% | 35% | ||
CT/seconds | MMPSO | 43.607 | 43.898 | 37.106 | 39.952 | 42.460 | 41.405 | 37.106 | |
MBPSO | 43.612 | 42.369 | 43.306 | 42.728 | 48.301 | 44.063 | 42.369 | ||
NSGA-Ⅱ | 45.038 | 48.282 | 44.382 | 45.383 | 46.448 | 45.907 | 44.382 | ||
MMPSO | 0.365 | 0.257 | 0.373 | 0.338 | 0.286 | 0.324 | 0.257 | ||
MBPSO | 0.412 | 0.379 | 0.408 | 0.292 | 0.238 | 0.346 | 0.238 | ||
NSGA-Ⅱ | 0.536 | 0.544 | 0.562 | 0.498 | 0.571 | 0.542 | 0.498 |
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