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## Green cross-dock based supply chain network design under demand uncertainty using new metaheuristic algorithms

 1 Department of management, Firoozkooh branch, Islamic Azad University, Firoozkooh, Iran 2 Department of Industrial Engineering and Future Studies, Faculty of Engineering, University of Isfahan, Iran 3 Department of Industrial Engineering, Qaemshahr branch, Islamic Azad University, Qaemshahr, Iran

* Corresponding author: Alireza Goli

Received  July 2020 Revised  March 2021 Early access June 2021

This study concerns the optimization of green supply chain network design under demand uncertainty. The issue of demand uncertainty has been addressed using a scenario-based analysis approach. The main contribution of this research is to investigate the optimization of cross-dock based supply chain under uncertainty using scenario-based formulation and metaheuristic algorithms. The problem has been formulated as a two-objective mathematical model with the objectives of minimizing the costs and minimizing the environmental impact of the supply chain. Two metaheuristic algorithms, namely non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective invasive weed optimization (MOIWO), have been developed to optimize this mathematical model. This paper focuses on the use of new metaheuristic algorithms such as MOIWO in green supply chain network design, which has received less attention in previous studies. The performance of the two solution methods has been evaluated in terms of three indices, which measure the quality, spacing, and diversification of solutions. Evaluations indicate that the developed MOIWO algorithm produces more Pareto solutions and solutions of higher quality than NSGA-II. A performance test carried out with 31 problem instances of different sizes shows that the two methods perform similarly in terms of the spread of solutions on the Pareto front, but MOIWO provides higher quality solutions than NSGA-II.

Citation: Arman Hamedirostami, Alireza Goli, Yousef Gholipour-Kanani. Green cross-dock based supply chain network design under demand uncertainty using new metaheuristic algorithms. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021105
##### References:
 [1] A. Abdi, N. Akbarpour, A. S. Amiri and M. Hajiaghaei-Keshteli, Innovative approaches to design and address green supply chain network with simultaneous pick-up and split delivery, Journal of Cleaner Production, 250 (2020), 119437. doi: 10.1016/j.jclepro.2019.119437.  Google Scholar [2] S. H. Amin, G. Zhang and M. Hajiaghaei-Keshteli, An integrated model for closed-loop supply chain configuration and supplier selection: Multi-objective approach, Expert Systems with Applications, 39 (2012), 6782-6791.  doi: 10.1016/j.eswa.2011.12.056.  Google Scholar [3] A. Baghalian, S. Rezapour and R. Z. Farahani, Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case, European Journal of Operational Research, 227 (2013), 199-215.  doi: 10.1016/j.ejor.2012.12.017.  Google Scholar [4] A. I. Barros, R. Dekker and V. Scholten, A two-level network for recycling sand: A case study, European Journal of Operational Research, 110 (1998), 199-214.  doi: 10.1016/S0377-2217(98)00093-9.  Google Scholar [5] T. F. Burgess, P. Grimshaw, L. H. Huatuco and N. E. Shaw, Mapping the operations and supply chain management field: A journal governance perspective, International Journal of Operations & Production Management, (2017). doi: 10.1108/IJOPM-01-2016-0043.  Google Scholar [6] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar [7] B. Fahimnia, J. Sarkis, F. Dehghanian, N. Banihashemi and S. Rahman, The impact of carbon pricing on a closed-loop supply chain: An Australian case study, Journal of Cleaner Production, 59 (2013), 210-225.  doi: 10.1016/j.jclepro.2013.06.056.  Google Scholar [8] R. Z. Farahani, S. Rezapour and L. Kardar, Supply Chain Sustainability and Raw Material Management: Concepts and Processes, (2012). doi: 10.4018/978-1-61350-504-5.  Google Scholar [9] Y.-H. Feng and G.-G. Wang, Binary moth search algorithm for discounted 0-1 knapsack problem, IEEE Access, 6 (2018), 10708-10719.  doi: 10.1109/ACCESS.2018.2809445.  Google Scholar [10] Y. Feng, S. Deb, G. G. Wang and A. H. Alavi, Monarch butterfly optimization: A comprehensive review, Expert Systems with Applications, (2020), 114418. doi: 10.1016/j.eswa.2020.114418.  Google Scholar [11] Y. Feng, G.-G. Wang, W. Li and N. Li, Multi-strategy monarch butterfly optimization algorithm for discounted 0-1 knapsack problem, Neural Computing and Applications, 30 (2018), 3019-3036.  doi: 10.1007/s00521-017-2903-1.  Google Scholar [12] Y. Feng, X. Yu and G.-G. Wang, A novel monarch butterfly optimization with global position updating operator for large-scale 0-1 Knapsack problems, Mathematics, 7 (2019), 1056. doi: 10.3390/math7111056.  Google Scholar [13] H. Garg, A hybrid PSO-GA algorithm for constrained optimization problems, Applied Mathematics and Computation, 274 (2016), 292-305.  doi: 10.1016/j.amc.2015.11.001.  Google Scholar [14] H. Garg, A hybrid GSA-GA algorithm for constrained optimization problems, Information Sciences, 478 (2019), 499-523.  doi: 10.1016/j.ins.2018.11.041.  Google Scholar [15] Z. Ghelichi, M. Saidi-Mehrabad and M. S. Pishvaee, A stochastic programming approach toward optimal design and planning of an integrated green biodiesel supply chain network under uncertainty: A case study, Energy, 156 (2018), 661-687.  doi: 10.1016/j.energy.2018.05.103.  Google Scholar [16] S. Gholipour, A. Ashoftehfard and H. Mina, Green supply chain network design considering inventory-location-routing problem: A fuzzy solution approach, International Journal of Logistics Systems and Management, 35 (2020), 436-452.  doi: 10.1504/IJLSM.2020.106272.  Google Scholar [17] H. Gholizadeh and H. Fazlollahtabar, Robust optimization and modified genetic algorithm for a closed loop green supply chain under uncertainty: Case study in melting industry, Computers & Industrial Engineering, 147 (2020), 106653. doi: 10.1016/j.cie.2020.106653.  Google Scholar [18] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadegheih, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.  doi: 10.1111/coin.12228.  Google Scholar [19] K. Govindan, H. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future, European Journal of Operational Research, 240 (2015), 603-626.  doi: 10.1016/j.ejor.2014.07.012.  Google Scholar [20] V. D. R. Guide, T. P. Harrison and L. N. Van Wassenhove, The challenge of closed-loop supply chains, Interfaces, 33 (2003), 3-6.  doi: 10.1287/inte.33.6.3.25182.  Google Scholar [21] H. C. Jang, Y. N. Lien and T. C. Tsai, Rescue information system for earthquake disasters based on MANET emergency communication platform. In Proceedings of the 2009 International Conference on Wireless Communications and Mobile Computing: Connecting the World Wirelessly, ACM, (2009), 623–627. doi: 10.1145/1582379.1582514.  Google Scholar [22] V. Jayaraman, V. D. R. Guide Jr and R. Srivastava, A closed-loop logistics model for remanufacturing, Journal of the Operational Research Society, (1999), 497–508. doi: 10.1057/palgrave.jors.2600716.  Google Scholar [23] G. Kannan, P. Sasikumar and K. Devika, A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling, Applied Mathematical Modelling, 34 (2010), 655-670.  doi: 10.1016/j.apm.2009.06.021.  Google Scholar [24] N. Lamba and P. Thareja, Developing the structural model based on analyzing the relationship between the barriers of green supply chain management using TOPSIS approach, Materials Today: Proceedings, (2020). Google Scholar [25] J. Li, H. Lei, A. H. Alavi and G.-G. Wang, Elephant herding optimization: variants, hybrids, and applications, Mathematics, 8 (2020), 1415. doi: 10.3390/math8091415.  Google Scholar [26] L. Liang and H. J. Quesada, Green design of a cellulosic butanol supply chain network: A case study of sorghum stem bio-butanol in missouri, BioResources, 13 (2018), 5617-5642.  doi: 10.15376/biores.13.3.5617-5642.  Google Scholar [27] R. Lotfi, Y. Z. Mehrjerdi, M. S. Pishvaee, A. Sadeghieh and G.-W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221-–253. doi: 10.3934/naco.2020023.  Google Scholar [28] D. Louwers, B. J. Kip, E. Peters, F. Souren, S. Douwe and P. Flapper, A facility location allocation model for reusing carpet materials, Computers & Industrial Engineering, 36 (1919), 855-869.  doi: 10.1016/S0360-8352(99)00168-0.  Google Scholar [29] A. R. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366.  doi: 10.1016/j.ecoinf.2006.07.003.  Google Scholar [30] S. Niroomand, H. Garg and A. Mahmoodirad, An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA Transactions, 107 (2020), 117-133.  doi: 10.1016/j.isatra.2020.07.033.  Google Scholar [31] R. S. Rad and N. Nahavandi, A novel multi-objective optimization model for integrated problem of green closed loop supply chain network design and quantity discount, Applied Mathematical Modelling, (2018). Google Scholar [32] A. Shabani, R. F. Saen and S. M. R. Torabipour, A new benchmarking approach in Cold Chain, Appl. Math. Model., 36 (2012), 212-224.  doi: 10.1016/j.apm.2011.05.051.  Google Scholar [33] T. Spengler, H. Püchert, T. Penkuhn and O. Rentz, Environmental integrated production and recycling management, In Produktion und Umwelt, Springer Berlin Heidelberg, (1997), 239–257. Google Scholar [34] M. Talaei, B. F. Moghaddam, M. S. Pishvaee, A. Bozorgi-Amiri and S. Gholamnejad, A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: A numerical illustration in electronics industry, Journal of Cleaner Production, 113 (2016), 662-673.  doi: 10.1016/j.jclepro.2015.10.074.  Google Scholar [35] M.-L. Tseng, M. S. Islam, N. Karia and F. Ahmad, A literature review on green supply chain management: Trends and future challenges, Resources, Conservation and Recycling, 141, (2019), 145–162. doi: 10.1016/j.resconrec.2018.10.009.  Google Scholar [36] G. G. Wang, Moth search algorithm: A bio-inspired metaheuristic algorithm for global optimization problems, Memetic Computing, 10, (2018), 151–164. Google Scholar [37] G.-G. Wang, S. Deb and L. D. S. Coelho, Elephant herding optimization, In 2015 3rd International Symposium on Computational and Business Intelligence (ISCBI), (2015), 1–5. doi: 10.1109/ISCBI.2015.8.  Google Scholar [38] G.-G. Wang, S. Deb and L. D. S. Coelho, Earthworm optimization algorithm: A bio-inspired metaheuristic algorithm for global optimization problems, International Journal of Bio-Inspired Computation, 12, (2018), 1–22. doi: 10.1504/IJBIC.2018.093328.  Google Scholar [39] G.-G. Wang, S. Deb and Z. Cui, Monarch butterfly optimization, Neural computing and applications, 31, (2019), 1995–2014. Google Scholar [40] G.-G. Wang, S. Deb, X.-Z. Gao and L. D. S. Coelho, A new metaheuristic optimization algorithm motivated by elephant herding behavior, International Journal of Bio-Inspired Computation, 8, (2016), 394–409. doi: 10.1504/IJBIC.2016.10002274.  Google Scholar [41] W. Xing, S. Y. Wang, Q. H. Zhao and G. W. Hua, Impact of fairness on strategies in dual-channel supply chain, Systems Engineering-Theory & Practice, 31, (2011), 1249–1256. Google Scholar [42] Q. Zhu, J. Sarkis and K.-H. Lai, Green supply chain management implications for "closing the loop", Transportation Research Part E: Logistics and Transportation Review, 44, (2008), 1–18. doi: 10.1016/j.tre.2006.06.003.  Google Scholar [43] E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transactions on Evolutionary Computation, 3, (1999), 257–271. doi: 10.1109/4235.797969.  Google Scholar

show all references

##### References:
 [1] A. Abdi, N. Akbarpour, A. S. Amiri and M. Hajiaghaei-Keshteli, Innovative approaches to design and address green supply chain network with simultaneous pick-up and split delivery, Journal of Cleaner Production, 250 (2020), 119437. doi: 10.1016/j.jclepro.2019.119437.  Google Scholar [2] S. H. Amin, G. Zhang and M. Hajiaghaei-Keshteli, An integrated model for closed-loop supply chain configuration and supplier selection: Multi-objective approach, Expert Systems with Applications, 39 (2012), 6782-6791.  doi: 10.1016/j.eswa.2011.12.056.  Google Scholar [3] A. Baghalian, S. Rezapour and R. Z. Farahani, Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case, European Journal of Operational Research, 227 (2013), 199-215.  doi: 10.1016/j.ejor.2012.12.017.  Google Scholar [4] A. I. Barros, R. Dekker and V. Scholten, A two-level network for recycling sand: A case study, European Journal of Operational Research, 110 (1998), 199-214.  doi: 10.1016/S0377-2217(98)00093-9.  Google Scholar [5] T. F. Burgess, P. Grimshaw, L. H. Huatuco and N. E. Shaw, Mapping the operations and supply chain management field: A journal governance perspective, International Journal of Operations & Production Management, (2017). doi: 10.1108/IJOPM-01-2016-0043.  Google Scholar [6] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar [7] B. Fahimnia, J. Sarkis, F. Dehghanian, N. Banihashemi and S. Rahman, The impact of carbon pricing on a closed-loop supply chain: An Australian case study, Journal of Cleaner Production, 59 (2013), 210-225.  doi: 10.1016/j.jclepro.2013.06.056.  Google Scholar [8] R. Z. Farahani, S. Rezapour and L. Kardar, Supply Chain Sustainability and Raw Material Management: Concepts and Processes, (2012). doi: 10.4018/978-1-61350-504-5.  Google Scholar [9] Y.-H. Feng and G.-G. Wang, Binary moth search algorithm for discounted 0-1 knapsack problem, IEEE Access, 6 (2018), 10708-10719.  doi: 10.1109/ACCESS.2018.2809445.  Google Scholar [10] Y. Feng, S. Deb, G. G. Wang and A. H. Alavi, Monarch butterfly optimization: A comprehensive review, Expert Systems with Applications, (2020), 114418. doi: 10.1016/j.eswa.2020.114418.  Google Scholar [11] Y. Feng, G.-G. Wang, W. Li and N. Li, Multi-strategy monarch butterfly optimization algorithm for discounted 0-1 knapsack problem, Neural Computing and Applications, 30 (2018), 3019-3036.  doi: 10.1007/s00521-017-2903-1.  Google Scholar [12] Y. Feng, X. Yu and G.-G. Wang, A novel monarch butterfly optimization with global position updating operator for large-scale 0-1 Knapsack problems, Mathematics, 7 (2019), 1056. doi: 10.3390/math7111056.  Google Scholar [13] H. Garg, A hybrid PSO-GA algorithm for constrained optimization problems, Applied Mathematics and Computation, 274 (2016), 292-305.  doi: 10.1016/j.amc.2015.11.001.  Google Scholar [14] H. Garg, A hybrid GSA-GA algorithm for constrained optimization problems, Information Sciences, 478 (2019), 499-523.  doi: 10.1016/j.ins.2018.11.041.  Google Scholar [15] Z. Ghelichi, M. Saidi-Mehrabad and M. S. Pishvaee, A stochastic programming approach toward optimal design and planning of an integrated green biodiesel supply chain network under uncertainty: A case study, Energy, 156 (2018), 661-687.  doi: 10.1016/j.energy.2018.05.103.  Google Scholar [16] S. Gholipour, A. Ashoftehfard and H. Mina, Green supply chain network design considering inventory-location-routing problem: A fuzzy solution approach, International Journal of Logistics Systems and Management, 35 (2020), 436-452.  doi: 10.1504/IJLSM.2020.106272.  Google Scholar [17] H. Gholizadeh and H. Fazlollahtabar, Robust optimization and modified genetic algorithm for a closed loop green supply chain under uncertainty: Case study in melting industry, Computers & Industrial Engineering, 147 (2020), 106653. doi: 10.1016/j.cie.2020.106653.  Google Scholar [18] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadegheih, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34.  doi: 10.1111/coin.12228.  Google Scholar [19] K. Govindan, H. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future, European Journal of Operational Research, 240 (2015), 603-626.  doi: 10.1016/j.ejor.2014.07.012.  Google Scholar [20] V. D. R. Guide, T. P. Harrison and L. N. Van Wassenhove, The challenge of closed-loop supply chains, Interfaces, 33 (2003), 3-6.  doi: 10.1287/inte.33.6.3.25182.  Google Scholar [21] H. C. Jang, Y. N. Lien and T. C. Tsai, Rescue information system for earthquake disasters based on MANET emergency communication platform. In Proceedings of the 2009 International Conference on Wireless Communications and Mobile Computing: Connecting the World Wirelessly, ACM, (2009), 623–627. doi: 10.1145/1582379.1582514.  Google Scholar [22] V. Jayaraman, V. D. R. Guide Jr and R. Srivastava, A closed-loop logistics model for remanufacturing, Journal of the Operational Research Society, (1999), 497–508. doi: 10.1057/palgrave.jors.2600716.  Google Scholar [23] G. Kannan, P. Sasikumar and K. Devika, A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling, Applied Mathematical Modelling, 34 (2010), 655-670.  doi: 10.1016/j.apm.2009.06.021.  Google Scholar [24] N. Lamba and P. Thareja, Developing the structural model based on analyzing the relationship between the barriers of green supply chain management using TOPSIS approach, Materials Today: Proceedings, (2020). Google Scholar [25] J. Li, H. Lei, A. H. Alavi and G.-G. Wang, Elephant herding optimization: variants, hybrids, and applications, Mathematics, 8 (2020), 1415. doi: 10.3390/math8091415.  Google Scholar [26] L. Liang and H. J. Quesada, Green design of a cellulosic butanol supply chain network: A case study of sorghum stem bio-butanol in missouri, BioResources, 13 (2018), 5617-5642.  doi: 10.15376/biores.13.3.5617-5642.  Google Scholar [27] R. Lotfi, Y. Z. Mehrjerdi, M. S. Pishvaee, A. Sadeghieh and G.-W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221-–253. doi: 10.3934/naco.2020023.  Google Scholar [28] D. Louwers, B. J. Kip, E. Peters, F. Souren, S. Douwe and P. Flapper, A facility location allocation model for reusing carpet materials, Computers & Industrial Engineering, 36 (1919), 855-869.  doi: 10.1016/S0360-8352(99)00168-0.  Google Scholar [29] A. R. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366.  doi: 10.1016/j.ecoinf.2006.07.003.  Google Scholar [30] S. Niroomand, H. Garg and A. Mahmoodirad, An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA Transactions, 107 (2020), 117-133.  doi: 10.1016/j.isatra.2020.07.033.  Google Scholar [31] R. S. Rad and N. Nahavandi, A novel multi-objective optimization model for integrated problem of green closed loop supply chain network design and quantity discount, Applied Mathematical Modelling, (2018). Google Scholar [32] A. Shabani, R. F. Saen and S. M. R. Torabipour, A new benchmarking approach in Cold Chain, Appl. Math. Model., 36 (2012), 212-224.  doi: 10.1016/j.apm.2011.05.051.  Google Scholar [33] T. Spengler, H. Püchert, T. Penkuhn and O. Rentz, Environmental integrated production and recycling management, In Produktion und Umwelt, Springer Berlin Heidelberg, (1997), 239–257. Google Scholar [34] M. Talaei, B. F. Moghaddam, M. S. Pishvaee, A. Bozorgi-Amiri and S. Gholamnejad, A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: A numerical illustration in electronics industry, Journal of Cleaner Production, 113 (2016), 662-673.  doi: 10.1016/j.jclepro.2015.10.074.  Google Scholar [35] M.-L. Tseng, M. S. Islam, N. Karia and F. Ahmad, A literature review on green supply chain management: Trends and future challenges, Resources, Conservation and Recycling, 141, (2019), 145–162. doi: 10.1016/j.resconrec.2018.10.009.  Google Scholar [36] G. G. Wang, Moth search algorithm: A bio-inspired metaheuristic algorithm for global optimization problems, Memetic Computing, 10, (2018), 151–164. Google Scholar [37] G.-G. Wang, S. Deb and L. D. S. Coelho, Elephant herding optimization, In 2015 3rd International Symposium on Computational and Business Intelligence (ISCBI), (2015), 1–5. doi: 10.1109/ISCBI.2015.8.  Google Scholar [38] G.-G. Wang, S. Deb and L. D. S. Coelho, Earthworm optimization algorithm: A bio-inspired metaheuristic algorithm for global optimization problems, International Journal of Bio-Inspired Computation, 12, (2018), 1–22. doi: 10.1504/IJBIC.2018.093328.  Google Scholar [39] G.-G. Wang, S. Deb and Z. Cui, Monarch butterfly optimization, Neural computing and applications, 31, (2019), 1995–2014. Google Scholar [40] G.-G. Wang, S. Deb, X.-Z. Gao and L. D. S. Coelho, A new metaheuristic optimization algorithm motivated by elephant herding behavior, International Journal of Bio-Inspired Computation, 8, (2016), 394–409. doi: 10.1504/IJBIC.2016.10002274.  Google Scholar [41] W. Xing, S. Y. Wang, Q. H. Zhao and G. W. Hua, Impact of fairness on strategies in dual-channel supply chain, Systems Engineering-Theory & Practice, 31, (2011), 1249–1256. Google Scholar [42] Q. Zhu, J. Sarkis and K.-H. Lai, Green supply chain management implications for "closing the loop", Transportation Research Part E: Logistics and Transportation Review, 44, (2008), 1–18. doi: 10.1016/j.tre.2006.06.003.  Google Scholar [43] E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transactions on Evolutionary Computation, 3, (1999), 257–271. doi: 10.1109/4235.797969.  Google Scholar
The structure of the studded supply chain
Validation of the mathematical model
Comparison of the algorithms in terms of MID
Comparison of the algorithms in terms of DM
Comparison of the algorithms in terms of SNS
Comparison of the algorithms in terms of solution time
The trends of Pareto solutions
Summary of the most notable articles in the field of green supply chain
 Authors Year of publication Supply chain network design Green supply chain Reducing costs Reducing environmental impacts Qualitative analysis Qualitative analysis Uncertainty Metaheuristic algorithms Barros et al. 1998 * * * - Louwers et al. 1999 * * * - Jayaraman et al. 1999 * * * - Guide et al. 2003 * * * * - Zhu et al. 2008 * * * * - Kannan et al. 2010 * * * * - Xing et al. 2011 * * * - Farahani et al. 2011 * * * - Shabani et al. 2021 * * * - Amin and Zhang 2012 * * * * * - Fahimnia et al. 2013 * * * - Talaei et al. 2016 * * * * * - Garb et al. 2016 * * PSO-GA Burgess et al. 2017 * * * * - Rad & Nahavandi 2018 * * * * - Ghelichi et al. 2018 * * * * * - Liang & Quesada 2018 * * - Goli et al. 2019 * * * IWO Garb et al. 2019 * * GSA-GA Lotfi et al. 2019 * * * * * - Niroomand et al. 2020 * * * * - Lamba & therja 2020 * * - Gholizadeh & Fazlollahtabar 2020 * * * GA Golipout et al. 2020 * * * * - Abdi et al. 2020 * * * * GA-PSO, RDA Present study 2020 * * * * * * MOIWO, NSGA-II
 Authors Year of publication Supply chain network design Green supply chain Reducing costs Reducing environmental impacts Qualitative analysis Qualitative analysis Uncertainty Metaheuristic algorithms Barros et al. 1998 * * * - Louwers et al. 1999 * * * - Jayaraman et al. 1999 * * * - Guide et al. 2003 * * * * - Zhu et al. 2008 * * * * - Kannan et al. 2010 * * * * - Xing et al. 2011 * * * - Farahani et al. 2011 * * * - Shabani et al. 2021 * * * - Amin and Zhang 2012 * * * * * - Fahimnia et al. 2013 * * * - Talaei et al. 2016 * * * * * - Garb et al. 2016 * * PSO-GA Burgess et al. 2017 * * * * - Rad & Nahavandi 2018 * * * * - Ghelichi et al. 2018 * * * * * - Liang & Quesada 2018 * * - Goli et al. 2019 * * * IWO Garb et al. 2019 * * GSA-GA Lotfi et al. 2019 * * * * * - Niroomand et al. 2020 * * * * - Lamba & therja 2020 * * - Gholizadeh & Fazlollahtabar 2020 * * * GA Golipout et al. 2020 * * * * - Abdi et al. 2020 * * * * GA-PSO, RDA Present study 2020 * * * * * * MOIWO, NSGA-II
Information of the problem used for validation
 Indices Symbol Value Total number of potential suppliers $|S|$ 15 Number of potential manufacturing plants $|M|$ 10 Number of potential distribution centers $|D|$ 20 Number of potential cross-docking warehouses $|C|$ 35 Number of retailers $|R|$ 50 Number of manufacturing technologies $|T|$ 3 Number of demand scenarios $KS|$ 3
 Indices Symbol Value Total number of potential suppliers $|S|$ 15 Number of potential manufacturing plants $|M|$ 10 Number of potential distribution centers $|D|$ 20 Number of potential cross-docking warehouses $|C|$ 35 Number of retailers $|R|$ 50 Number of manufacturing technologies $|T|$ 3 Number of demand scenarios $KS|$ 3
Parameter setting used for the validation problem
 Parameter Description Value $TEC_{pt}$ The cost of using the technology $t$ in the plant $p$ 15000 $OC_{p}$ The cost of opening the plant $p$ 60000 $OC_{d}$ The cost of opening the distribution center $d$ 20000 $OC_{c}$ The cost of opening the cross-docking warehouse $c$ 7000 $FC_{s_g}$ The fixed cost of a long-term relationship with the supplier $S_g$ 1000 $VC_{d}$ The variable cost of moving the product in the distribution center $d$ 100 $VC_{c}$ The variable cost of moving the product in the cross-docking warehouse $c$ 80 $TC_{s_gp}$ The cost of transporting each unit of product from the supplier $S_g$ to the plant $p$ 45 $TC_{pd}$ The cost of transporting each unit of product from the plant $p$ to the distribution center $d$ 50 $TC_{dc}$ The cost of transporting each unit of product from the distribution center $d$ to the cross-docking warehouse $c$ 30 $TC_{cr}$ The cost of transporting each unit of product from the cross-docking warehouse $c$ to the retailer $r$ 25 $TC_{dr}$ The cost of transporting each unit of product from the distribution center $d$ to the retailer $r$ 30 $MC_{pt}$ The cost of manufacturing each unit of product in the plant $p$ using the technology $t$ 190 $EM_{pt}$ The greenhouse gas emission cost due to the manufacturing of each unit of product in the plant $p$ using the technology $t$ 230 $ET_{s_gp}$ The environmental impact of transporting each unit of product from the supplier $S_g$ to the plant $p$ 70 $ET_{pd}$ The environmental impact of transporting each unit of product from the plant $p$ to the distribution center $d$ 60 $ET_{dc}$ The environmental impact of transporting each unit of product from the distribution center $d$ to the cross-docking warehouse $c$ 60 $ET_{cr}$ The environmental impact of transporting each unit of product from the cross-docking warehouse $c$ to the retailer $r$ 60 $ET_{dr}$ The environmental impact of transporting each unit of product from the distribution center $d$ to the retailer $r$ 40 $ES_{s_g}$ The environmental impact of the supplier $S_g$ 130 $EO_{pt}$ The environmental impact of opening the plant $p$ using the technology $t$ 230 $EO_{d}$ The environmental impact of opening the distribution center $d$ 160 $EO_{c}$ The environmental impact of opening the cross-docking warehouse $c$ 150 $Cap_{s_g}$ Capacity of the supplier $S_g$ 25000 $Cap_{pt}$ Production capacity of the plant $p$ using technology $t$ 20000 $Cap_{d}$ Product transfer capacity at the distribution center $d$ 15000 $Cap_{c}$ Product transfer capacity at in the cross-docking warehouse $c$ 10000 $S_{Max}$ The maximum number of suppliers required 6 $P_{Max}$ The maximum number of plants required 5 $D_{Max}$ The maximum number of distribution centers required 10 $C_{Max}$ The maximum number of cross-docking warehouses required 30 $P(\xi)$ The probability of occurrence of each scenario 0.1-0.3-0.6
 Parameter Description Value $TEC_{pt}$ The cost of using the technology $t$ in the plant $p$ 15000 $OC_{p}$ The cost of opening the plant $p$ 60000 $OC_{d}$ The cost of opening the distribution center $d$ 20000 $OC_{c}$ The cost of opening the cross-docking warehouse $c$ 7000 $FC_{s_g}$ The fixed cost of a long-term relationship with the supplier $S_g$ 1000 $VC_{d}$ The variable cost of moving the product in the distribution center $d$ 100 $VC_{c}$ The variable cost of moving the product in the cross-docking warehouse $c$ 80 $TC_{s_gp}$ The cost of transporting each unit of product from the supplier $S_g$ to the plant $p$ 45 $TC_{pd}$ The cost of transporting each unit of product from the plant $p$ to the distribution center $d$ 50 $TC_{dc}$ The cost of transporting each unit of product from the distribution center $d$ to the cross-docking warehouse $c$ 30 $TC_{cr}$ The cost of transporting each unit of product from the cross-docking warehouse $c$ to the retailer $r$ 25 $TC_{dr}$ The cost of transporting each unit of product from the distribution center $d$ to the retailer $r$ 30 $MC_{pt}$ The cost of manufacturing each unit of product in the plant $p$ using the technology $t$ 190 $EM_{pt}$ The greenhouse gas emission cost due to the manufacturing of each unit of product in the plant $p$ using the technology $t$ 230 $ET_{s_gp}$ The environmental impact of transporting each unit of product from the supplier $S_g$ to the plant $p$ 70 $ET_{pd}$ The environmental impact of transporting each unit of product from the plant $p$ to the distribution center $d$ 60 $ET_{dc}$ The environmental impact of transporting each unit of product from the distribution center $d$ to the cross-docking warehouse $c$ 60 $ET_{cr}$ The environmental impact of transporting each unit of product from the cross-docking warehouse $c$ to the retailer $r$ 60 $ET_{dr}$ The environmental impact of transporting each unit of product from the distribution center $d$ to the retailer $r$ 40 $ES_{s_g}$ The environmental impact of the supplier $S_g$ 130 $EO_{pt}$ The environmental impact of opening the plant $p$ using the technology $t$ 230 $EO_{d}$ The environmental impact of opening the distribution center $d$ 160 $EO_{c}$ The environmental impact of opening the cross-docking warehouse $c$ 150 $Cap_{s_g}$ Capacity of the supplier $S_g$ 25000 $Cap_{pt}$ Production capacity of the plant $p$ using technology $t$ 20000 $Cap_{d}$ Product transfer capacity at the distribution center $d$ 15000 $Cap_{c}$ Product transfer capacity at in the cross-docking warehouse $c$ 10000 $S_{Max}$ The maximum number of suppliers required 6 $P_{Max}$ The maximum number of plants required 5 $D_{Max}$ The maximum number of distribution centers required 10 $C_{Max}$ The maximum number of cross-docking warehouses required 30 $P(\xi)$ The probability of occurrence of each scenario 0.1-0.3-0.6
Demand information in the scenarios
 Scenario No. 1 2 3 Probability of senario 0.1 0.6 0.3 Retailer demand 100 250 320
 Scenario No. 1 2 3 Probability of senario 0.1 0.6 0.3 Retailer demand 100 250 320
The selected suppliers in the optimal solution
 Scenario No. 2 3 4 5
 Scenario No. 2 3 4 5
The selected plants and manufacturing technologies in the optimal solution
 Plant No. 1 5 6 7 9 Technology ID 1 1 2 3 2
 Plant No. 1 5 6 7 9 Technology ID 1 1 2 3 2
The distribution centers chosen in the optimal solution
 Distribution center No. 7 9 14 15 18 19 20
 Distribution center No. 7 9 14 15 18 19 20
The cross-docking warehouses chosen in the optimal solution
 Cross-docking warehouse No. 2 4 9 10 11 14 18 19 26 28 29 30
 Cross-docking warehouse No. 2 4 9 10 11 14 18 19 26 28 29 30
Details of the output of the mathematical model for the validation problem
 Items Cost Environmental impact Selecting suppliers 4000 520 Establishing manufacturing plants 300000 75000 Establishing distribution centers 160000 1280 Establishing warehouses 84000 1800 Manufacturing the productb 37000 960 Transportation 34000 1620 Total 619000 81180
 Items Cost Environmental impact Selecting suppliers 4000 520 Establishing manufacturing plants 300000 75000 Establishing distribution centers 160000 1280 Establishing warehouses 84000 1800 Manufacturing the productb 37000 960 Transportation 34000 1620 Total 619000 81180
Information of the numerical problem instances used for performance evaluation
 Instance No. $|R|$ $|D|$ $|C|$ $|M|$ $|S|$ $|T|$ $|KS|$ Instance No. $|R|$ $|D|$ $|C|$ $|M|$ $|S|$ $|T|$ $|KS|$ 1 6 4 2 1 1 1 2 2 8 5 2 1 2 1 2 3 10 5 2 1 2 1 3 4 12 7 2 1 3 2 3 5 14 8 3 1 3 2 4 6 16 8 3 1 4 2 4 7 18 10 3 2 4 3 5 8 20 12 3 2 5 3 5 9 22 12 4 2 5 3 6 10 24 15 4 2 6 4 6 11 26 16 4 2 6 4 7 12 28 18 5 2 7 4 7 13 30 20 6 3 7 5 8 14 35 24 7 3 8 5 8 15 40 25 8 3 8 5 9 16 45 25 9 3 9 6 9 17 50 30 10 4 9 6 10 18 55 30 11 4 10 6 10 19 60 40 12 4 10 7 11 20 65 40 13 5 11 7 11 21 70 50 14 5 11 7 12 22 75 50 15 5 12 8 12 23 80 60 16 6 12 8 13 24 85 60 17 6 13 8 13 25 90 70 18 6 13 9 14 26 95 70 19 7 14 9 14 27 100 80 20 7 14 9 15 28 105 80 21 7 15 10 15 29 110 90 22 8 15 10 16 30 115 90 23 8 16 10 16 31 120 100 24 8 16 11 17
 Instance No. $|R|$ $|D|$ $|C|$ $|M|$ $|S|$ $|T|$ $|KS|$ Instance No. $|R|$ $|D|$ $|C|$ $|M|$ $|S|$ $|T|$ $|KS|$ 1 6 4 2 1 1 1 2 2 8 5 2 1 2 1 2 3 10 5 2 1 2 1 3 4 12 7 2 1 3 2 3 5 14 8 3 1 3 2 4 6 16 8 3 1 4 2 4 7 18 10 3 2 4 3 5 8 20 12 3 2 5 3 5 9 22 12 4 2 5 3 6 10 24 15 4 2 6 4 6 11 26 16 4 2 6 4 7 12 28 18 5 2 7 4 7 13 30 20 6 3 7 5 8 14 35 24 7 3 8 5 8 15 40 25 8 3 8 5 9 16 45 25 9 3 9 6 9 17 50 30 10 4 9 6 10 18 55 30 11 4 10 6 10 19 60 40 12 4 10 7 11 20 65 40 13 5 11 7 11 21 70 50 14 5 11 7 12 22 75 50 15 5 12 8 12 23 80 60 16 6 12 8 13 24 85 60 17 6 13 8 13 25 90 70 18 6 13 9 14 26 95 70 19 7 14 9 14 27 100 80 20 7 14 9 15 28 105 80 21 7 15 10 15 29 110 90 22 8 15 10 16 30 115 90 23 8 16 10 16 31 120 100 24 8 16 11 17
Performance evaluation of NSGA II and MOIWO
 NSGA II MOIWO Test problem MID DM SNS Solution time MID DM SNS Solution time 1 2128.402 388.3026 337.138 19.6 2392.871 387.0695 62.36729 21.9 2 9901.841 947.1654 1327.495 24.8 10025.83 2126.98 811.6118 22.2 3 14960.24 1626.795 1424.479 26.9 17064.71 1115.375 630.3547 23.5 4 26614.19 656.5366 2013.4 34.8 29887.93 71.44966 36.73556 34.4 5 43885.55 3292.813 2982.944 39.7 43253.99 1438.363 1676.061 38.6 6 65925.99 1670.296 3692.972 45.6 65007.11 3411.264 2778.861 42.3 7 170150.2 7986.59 6394.361 51.7 172745.8 3498.732 2442.37 55.3 8 252032.8 5583.598 4870.001 62.7 256509.7 5928.794 4474.071 57.5 9 284951.5 16779.53 9177.958 63.8 273177.9 6887.26 4848.212 65.1 10 381924 15844.87 4170.522 67.4 367442.1 8144.148 10614.28 74.2 11 407187.7 13023.62 5017.187 73.5 396215.3 3333.181 2116.804 76.5 12 511353.5 14114.78 6296.453 78.9 500211 5713.979 7539.665 79.9 13 564882.1 15743.56 5480.24 84.9 535189.2 10652.97 5908.288 87.8 14 1018173 16063.69 4831.171 95.1 537751 11154.74 6256.459 93.8 15 666245.3 19694.52 6667.181 102.1 565154 12045.07 7077.878 109.7 16 633000.8 12577.59 7022.577 116.3 590704.9 13867.63 8038.837 121.4 17 746779.6 12953.08 5979.238 121.9 636111.9 14579.36 8050.933 133.2 18 843457.6 13023.62 6144.048 127.3 711607.4 15010.54 9184.642 146.1 19 795303.2 14352.39 6745.26 138.6 715590.4 16567.74 10596.68 150.5 20 948178.2 17867.75 7016.65 144.8 840491.9 18883.24 11367.66 150.9 21 889252.7 18674.4 5709.634 152.6 843119.3 19923.22 11808.66 153.6 22 979589.8 18340.73 6096.018 172.8 964187.6 22824.67 12901.56 162.4 23 1252267 22662.02 5907.127 189.6 1124600 25030.35 13380.75 184.6 24 1399254 24910.39 6013.365 205.4 1346977 25886.89 14847.5 209.6 25 1626187 26719.46 5702.392 232.2 1431072 27008.7 15218.09 218.6 26 1808562 27314.07 6976.036 237.2 1551875 32284.12 16639.52 233.6 27 2165170 26285.52 6046.077 249.8 1835831 32571.71 17894.87 262.8 28 2115911 34620.55 6021.699 278.2 2071615 37901.51 20138.68 289.8 29 2601260 31453.13 5924.462 286.4 2418931 41658.63 23395.58 290.2 30 3037899 40728.54 6957.946 328.4 2731168 47049.29 27300.69 303.6 31 3450494 34538.37 7073.724 359.8 2986884 47585.72 31641.19 336.2 Average 958480.1 16465.75 5355.476 135.9 857186.9 16598.15 9989.673 136.4
 NSGA II MOIWO Test problem MID DM SNS Solution time MID DM SNS Solution time 1 2128.402 388.3026 337.138 19.6 2392.871 387.0695 62.36729 21.9 2 9901.841 947.1654 1327.495 24.8 10025.83 2126.98 811.6118 22.2 3 14960.24 1626.795 1424.479 26.9 17064.71 1115.375 630.3547 23.5 4 26614.19 656.5366 2013.4 34.8 29887.93 71.44966 36.73556 34.4 5 43885.55 3292.813 2982.944 39.7 43253.99 1438.363 1676.061 38.6 6 65925.99 1670.296 3692.972 45.6 65007.11 3411.264 2778.861 42.3 7 170150.2 7986.59 6394.361 51.7 172745.8 3498.732 2442.37 55.3 8 252032.8 5583.598 4870.001 62.7 256509.7 5928.794 4474.071 57.5 9 284951.5 16779.53 9177.958 63.8 273177.9 6887.26 4848.212 65.1 10 381924 15844.87 4170.522 67.4 367442.1 8144.148 10614.28 74.2 11 407187.7 13023.62 5017.187 73.5 396215.3 3333.181 2116.804 76.5 12 511353.5 14114.78 6296.453 78.9 500211 5713.979 7539.665 79.9 13 564882.1 15743.56 5480.24 84.9 535189.2 10652.97 5908.288 87.8 14 1018173 16063.69 4831.171 95.1 537751 11154.74 6256.459 93.8 15 666245.3 19694.52 6667.181 102.1 565154 12045.07 7077.878 109.7 16 633000.8 12577.59 7022.577 116.3 590704.9 13867.63 8038.837 121.4 17 746779.6 12953.08 5979.238 121.9 636111.9 14579.36 8050.933 133.2 18 843457.6 13023.62 6144.048 127.3 711607.4 15010.54 9184.642 146.1 19 795303.2 14352.39 6745.26 138.6 715590.4 16567.74 10596.68 150.5 20 948178.2 17867.75 7016.65 144.8 840491.9 18883.24 11367.66 150.9 21 889252.7 18674.4 5709.634 152.6 843119.3 19923.22 11808.66 153.6 22 979589.8 18340.73 6096.018 172.8 964187.6 22824.67 12901.56 162.4 23 1252267 22662.02 5907.127 189.6 1124600 25030.35 13380.75 184.6 24 1399254 24910.39 6013.365 205.4 1346977 25886.89 14847.5 209.6 25 1626187 26719.46 5702.392 232.2 1431072 27008.7 15218.09 218.6 26 1808562 27314.07 6976.036 237.2 1551875 32284.12 16639.52 233.6 27 2165170 26285.52 6046.077 249.8 1835831 32571.71 17894.87 262.8 28 2115911 34620.55 6021.699 278.2 2071615 37901.51 20138.68 289.8 29 2601260 31453.13 5924.462 286.4 2418931 41658.63 23395.58 290.2 30 3037899 40728.54 6957.946 328.4 2731168 47049.29 27300.69 303.6 31 3450494 34538.37 7073.724 359.8 2986884 47585.72 31641.19 336.2 Average 958480.1 16465.75 5355.476 135.9 857186.9 16598.15 9989.673 136.4
Comparison of metaheuristic algorithms with random search
 Iteration MID DM SNS Random simulation 100 1347512 17541.19 3421.96 Random simulation 500 1296475 20047.48 4343.55 Random simulation 1000 1253954 22096.37 5736.47 MOIWO 200 1124600 25030.35 13380.75 NSGA-II 200 1252267 22662.02 5907.12 Best method - MOIWO MOIWO MOIWO
 Iteration MID DM SNS Random simulation 100 1347512 17541.19 3421.96 Random simulation 500 1296475 20047.48 4343.55 Random simulation 1000 1253954 22096.37 5736.47 MOIWO 200 1124600 25030.35 13380.75 NSGA-II 200 1252267 22662.02 5907.12 Best method - MOIWO MOIWO MOIWO
Results of the t-test on the MID values of the algorithms
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-MID MOIWO-MID 101293.165 135632.934 24360.394 51542.603 151043.727 4.158 30 .000
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-MID MOIWO-MID 101293.165 135632.934 24360.394 51542.603 151043.727 4.158 30 .000
Results of the t-test on the SNS values of the algorithms
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-SNS MOIWO-SNS -4634.19 7169.486 1287.677 -7263.98 -2004.40 -3.599 30 .001
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-SNS MOIWO-SNS -4634.19 7169.486 1287.677 -7263.98 -2004.40 -3.599 30 .001
Results of the t-test on the DM values of the algorithms
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-DM MOIWI-DM -132.401 5447.236 978.352 -2130.46 1865.66 -.135 30 .893
 Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference t df Sig. (2-tailed) Lower Upper NSGAII-DM MOIWI-DM -132.401 5447.236 978.352 -2130.46 1865.66 -.135 30 .893
2 Sensitivity analysis results
 $p$=0.8 $p$=0.9 $p$=0 $p$=1.1 $p$=1.2 OB1 OB2 OB1 OB2 OB1 OB2 OB1 OB2 OB1 OB2 575413 73608 606912 78914 619000 81180 621994 84252 630881 87539 596470 70029 619944 74532 627820 75855 655736 78734 699668 86211 605106 66585 661924 69282 690996 72564 703981 77182 795615 83357 616235 66387 759706 65312 730679 70027 728719 70189 819769 76894 639479 66268 797345 61378 751322 67136 880525 63570 876866 71727 702385 60188 826948 57410 809028 64373 896155 59665 972932 66554 775653 56490 948546 54882 879333 60945 934413 56740 983940 60876
 $p$=0.8 $p$=0.9 $p$=0 $p$=1.1 $p$=1.2 OB1 OB2 OB1 OB2 OB1 OB2 OB1 OB2 OB1 OB2 575413 73608 606912 78914 619000 81180 621994 84252 630881 87539 596470 70029 619944 74532 627820 75855 655736 78734 699668 86211 605106 66585 661924 69282 690996 72564 703981 77182 795615 83357 616235 66387 759706 65312 730679 70027 728719 70189 819769 76894 639479 66268 797345 61378 751322 67136 880525 63570 876866 71727 702385 60188 826948 57410 809028 64373 896155 59665 972932 66554 775653 56490 948546 54882 879333 60945 934413 56740 983940 60876
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