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July  2019, 15(3): 1263-1288. doi: 10.3934/jimo.2018095

Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization

Xia Zhao a,b,, and  Jianping Dou c,

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

* Corresponding author: Xia Zhao

Received  July 2017 Revised  January 2018 Published  July 2019 Early access  July 2018

Fund Project: This work is supported by the Programs of National Natural Science Foundation of China(No.51575108 and No.71403114), China Special Fund for Grain-scientific Research in the Public Interest(No.201513004), the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD) and Qinglan Project.

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.

Keywords: Supply chain network design, transportation choice, multi-objective optimization, particle swarm optimization.
Mathematics Subject Classification: 90B80.
Citation: Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095
References:
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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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[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.  Google Scholar

[12]

K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Kennedy and R. Eberhart, Particle swarm optimization, Piscataway, NJ: IEEE Service Center, 4 (1995), 1942-1948.  doi: 10.1109/ICNN.1995.488968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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Figure 1.  The studied single product four-echelon supply chain network
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Figure 2.  The solution representation of binary and continuous variables by a particle
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Figure 3.  Solution procedure of the MILP model by PSO and LINGO
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Figure 8.  Location map of an apple supply chain in Shanxi Province of China
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Figure 4.  Solution report of case 1 by LINGO given DFR as one
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Figure 5.  Non-dominated solutions of case 1 obtained by three algorithms
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Figure 6.  Non-dominated solutions of problem 4 obtained by three algorithms
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Figure 7.  Average and best values of ERs and ∆ versus nine problems
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Table 1.  Key decision variables of five typical non-dominated solutions for case 1
# 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
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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
Table 6.  Expected maximal sale amounts of fresh apples for nine CZs(K tons)
CZs BJ SH SZ WH CS ZZ CD CQ XA
Apples 30 40 20 40 50 30 40 30 50
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CZs BJ SH SZ WH CS ZZ CD CQ XA
Apples 30 40 20 40 50 30 40 30 50
Table 7.  Capacities and procumbent costs for five suppliers
Locations YA TC BO XY WN
Capacity (K tons) 120 60 60 160 100
cost (RMB/ton) 610 620 630 620 630
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Locations YA TC BO XY WN
Capacity (K tons) 120 60 60 160 100
cost (RMB/ton) 610 620 630 620 630
Table 8.  Variable costs of producing fresh apples for potential plants (RMB/ton)
Sites YA TC BO XY WN XA
Cost 500 520 540 530 530 520
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Sites YA TC BO XY WN XA
Cost 500 520 540 530 530 520
Table 9.  Variable costs of warehousing for nine sites (RMB/ton)
DCs BJ SH SZ WH CS ZZ CD CQ XA
Cost 420 430 440 410 400 400 400 400 400
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DCs BJ SH SZ WH CS ZZ CD CQ XA
Cost 420 430 440 410 400 400 400 400 400
Table 10.  Unit transportation costs for supplier-to-plant (RMB/ton)
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
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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
Table 11.  Unit transportation costs of truck and rail (RMB/ton)
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.
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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.
Table 5.  Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three medium and three large randomly generated problems
Pro. Item ER CT/seconds $\Delta$
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
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Pro. Item ER CT/seconds $\Delta$
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
Table 2.  Characteristics of all randomly generated SCND problems
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
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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
Table 3.  Comparison of location decisions for case 3 with and without 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
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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
Table 4.  Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three small cases
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 $45\%$ $15\%$
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
$\Delta$ 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
$\Delta$ 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
$\Delta$ 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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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 $45\%$ $15\%$
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
$\Delta$ 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
$\Delta$ 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
$\Delta$ 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
[1]

Min Zhang, Gang Li. Multi-objective optimization algorithm based on improved particle swarm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1413-1426. doi: 10.3934/dcdss.2019097
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