
-
Previous Article
Worst-case analysis of Gini mean difference safety measure
- JIMO Home
- This Issue
-
Next Article
Equilibrium strategies in a supply chain with capital constrained suppliers: The impact of external financing
A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints
1. | Faculty of Engineering, University of Kurdistan, Pasdaran Blvd., Post Box: 416, Sanandaj, Iran |
2. | MSC of Industrial Engineering, University of Kurdistan, Sanandaj, Iran |
The present study considers the transport discounts and capacity constraints for the suppliers and manufacturers simultaneously to provide a multi-objective decision-making model for supplier selection on a three-level supply chain. For this purpose, it begins with presenting a nonlinear mixed-integer model of the problem, where the objectives include the minimization of the logistics costs and lead time. Subsequently, the NSGA-Ⅱ algorithm is developed to solve the large-scale model of the problem and simultaneously optimize the two objectives to achieve Pareto-optimal solutions. To test the efficiency of the proposed algorithm, several synthetic examples of various sizes are then generated and solved. Finally, the paper compares the performance of the proposed metaheuristic algorithm with the augmented epsilon-constraint method. In summary, the findings of this study provided researchers and industries to easily access to a cohesive model of supplier selection considering transportation that are essential to the solution of many real-world challenging logistics issues.
References:
[1] |
A. Aguezzoul and P. Ladet, A nonlinear multiobjective approach for the supplier selection, integrating transportation policies, J. Model. Man., 2, (2007), 157–169.
doi: 10.1108/17465660710763434. |
[2] |
N. Aissaoui, M. Haouari and E. Hassini, Supplier selection and order lot sizing modeling: A review, Comput. Oper. Res., 34 (2007), no. 12, 3516–3540.
doi: 10.1016/j.cor.2006.01.016. |
[3] |
S. Amin and G. Zhang, An integrated model for closed-loop supply chain configuration and supplier selection: Multi-objective approach, Expert Syst. Appl., 39 (2012), no. 8, 6782–6791.
doi: 10.1016/j.eswa.2011.12.056. |
[4] |
T. F. Anthony and F. P. Buffa, Strategic purchase scheduling, J. Purch. Mater. Manag., 13 (1977), no. 3, 27–31.
doi: 10.1111/j.1745-493X.1977.tb00400.x. |
[5] |
F. Arikan, A fuzzy solution approach for multi objective supplier selection, Expert Syst. Appl., 40 (2013), no. 3,947–952.
doi: 10.1016/j.eswa.2012.05.051. |
[6] |
E. Babaee Tirkolaee, A. Goli, G.-W. Weber, Multi-Objective Aggregate Production Planning Model Considering Overtime and Outsourcing Options Under Fuzzy Seasonal Demand, in Adv. Manuf. II, Springer, 2019, 81–96. Google Scholar |
[7] |
E. Babaee Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G.-W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, J. of Cleaner Prod., 250 (2020), 119517.
doi: 10.1016/j.jclepro.2019.119517. |
[8] |
M. Bevilacqua, F. E. Ciarapica and G. Giacchetta, A fuzzy-QFD approach to supplier selection, J. Purchase Supp. Man., 12 (2006), no. 1, 14–27.
doi: 10.1016/j.pursup.2006.02.001. |
[9] |
E. Buzdogan-Lindenmayr, G. Kara and A. Selcuk-Kestel, Sevtap Assessment of supplier risk for copper procurement, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (2019), no. 1, 1045–1060.
doi: 10.31801/cfsuasmas.501491. |
[10] |
S. Chou and Y. Chang, A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach, Expert Syst. Appl., 34 (2008), no. 4, 2241–2253.
doi: 10.1016/j.eswa.2007.03.001. |
[11] |
K. Deb and A. Pratap, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Trans. E. Comput., 6 (2002), no. 2,182–197.
doi: 10.1109/4235.996017. |
[12] |
E. A. Demirtas and O. Ustun, Analytic network process and multi-period goal programming integration in purchasing decisions, Comput. Ind. Eng., 56 (2009), no. 2,677–690.
doi: 10.1016/j.cie.2006.12.006. |
[13] |
S. Deng, R. Aydin, C. K. K. Kwong and Y. Huang,
Integrated product line design and supplier selection: A multi-objective optimization paradigm, Comput. Ind. Eng., 70 (2014), 150-158.
doi: 10.1016/j.cie.2014.01.011. |
[14] |
R. M. Ebrahim, J. Razmi and H. Haleh, Scatter search algorithm for supplier selection and order lot sizing under multiple price discount environment, Adv. Eng. Softw., 40 (2009), no. 9,766–776.
doi: 10.1016/j.advengsoft.2009.02.003. |
[15] |
A. Ekici, An improved model for supplier selection under capacity constraint and multiple criteria, Int. J. Prod. Econ., 141 (2013), no. 2,574–581.
doi: 10.1016/j.ijpe.2012.09.013. |
[16] |
R. Farzipoor Sean, A new mathematical approach for suppliers selection: Accounting for non-homogeneity is important, Appl. Math. Comput., 185 (2007), no. 1, 84–95.
doi: 10.1016/j.amc.2006.07.071. |
[17] |
A. Gaballa, Minimum cost allocation of tenders, J. Oper. Res. Soc., 25 (1974), no. 3,389–398.
doi: 10.1057/jors.1974.73. |
[18] |
S. Ghodsypour and C. O$\mathop {\rm{B}}\limits^{'} $rien, The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint, Int. J. Prod. Econ., 73 (2001), no. 1, 15–27.
doi: 10.1016/S0925-5273(01)00093-7. |
[19] |
X. Hu and J. G. Motwani,
Minimizing downside risks for global sourcing under price-sensitive stochastic demand, exchange rate uncertainties and supplier capacity constraints, Int. J. Prod. Econ., 147 (2014), 398-409.
doi: 10.1016/j.ijpe.2013.04.045. |
[20] |
O. Jadidi, S. Zolfaghari and S. Cavalieri,
A new normalized goal programming model for multi-objective problems: A case of supplier selection and order allocation, Int. J. Prod. Econ., 148 (2014), 158-165.
doi: 10.1016/j.ijpe.2013.10.005. |
[21] |
R. Jazemi, J. Gheidar-Kheljani and S. H. Ghodsypour, Modeling the multiobjective problem of supplier selection, taking into account the benefits of the buyer and the suppliers simultaneously, IEEE Trans. Ind. Inform., 7 (2010), no. 3,517–526.
doi: 10.1109/TII.2011.2158835. |
[22] |
D. Kannan, R. Khodaverdi, L. Olfat, A. Jafarian and A. Diabat,
Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain, J. Clean. Prod., 47 (2013), 355-367.
doi: 10.1016/j.jclepro.2013.02.010. |
[23] |
M. Khosroabadi, M. Lotfi and H. Khademizare, Supplier selection program and sizing orders for products at a discounted price and the cost of transportation, J. of Ind. Eng. Res. in Prod. Sys., (2013), no. 1,139–153. Google Scholar |
[24] |
Z. Liao and J. Rittscher, A multi-objective supplier selection model under stochastic demand conditions, 105 (2007), 150–159
doi: 10.1016/j.ijpe.2006.03.001. |
[25] |
R. Lotfi, G.-W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Ind. Manag. Optim., 16 (2020), no. 1,117–140.
doi: 10.3934/jimo.2018143. |
[26] |
G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Appl. Math. Comput., 213 (2009), no. 2,455–465.
doi: 10.1016/j.amc.2009.03.037. |
[27] |
G. Mavrotas and K. Florios, An improved version of the augmented $\varepsilon$-constraint method (AUGMECON 2) for finding the exact pareto set in multi-objective integer programming problems, Appl. Math. Comput., 219 (2013), no. 18, 9652–9669.
doi: 10.1016/j.amc.2013.03.002. |
[28] |
A. Mendoza and J. A. Ventura, Analytical models for supplier selection and order quantity allocation, Appl. Math. Model., 36 (2012), no. 8, 3826–3835.
doi: 10.1016/j.apm.2011.11.025. |
[29] |
K. S. Moghaddam,
Fuzzy multi-objective model for supplier selection and order allocation in reverse logistics systems under supply and demand uncertainty, Expert Syst. Appl., 42 (2015), 6237-6254.
doi: 10.1016/j.eswa.2015.02.010. |
[30] |
D. Mohammaditabar, S. H. Ghodsypour and A. Hafezalkotob,
A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs, Int. J. Prod. Econ., 181 (2016), 87-97.
doi: 10.1016/j.ijpe.2015.11.016. |
[31] |
S. Nazari-Shirkouhi, H. Shakouri, B. Javadi and A. Keramati, Supplier selection and order allocation problem using a two-phase fuzzy multi-objective linear programming, Appl. Math. Model., 37 (2013), no. 22, 9308–9323.
doi: 10.1016/j.apm.2013.04.045. |
[32] |
A. Pan, Allocation of order quantity among suppliers, J. Purch. Mater. Manag., 25 (1989), no. 3, 36–39.
doi: 10.1111/j.1745-493X.1989.tb00489.x. |
[33] |
K. Shaw, R. Shankar, S. S. Yadav and L. S. Thakur, Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain, Expert Syst. Appl., 39 (2012), no. 9, 8182–8192.
doi: 10.1016/j.eswa.2012.01.149. |
[34] |
Y.-C. Tsao and J.-C. Lu, A supply chain network design considering transportation cost discounts, Transp. Res. Part E Logist. Transp. Rev., 48 (2012), no. 2,401–414.
doi: 10.1016/j.tre.2011.10.004. |
[35] |
C. A. Weber, J. R. Current and W. C. Benton, Vendor selection criteria and methods, Eur. J. Oper. Res., 50 (1991), no. 1, 2–18.
doi: 10.1016/0377-2217(91)90033-R. |
[36] |
W. Xia and Z. Wu, Supplier selection with multiple criteria in volume discount environments, Omega, 35 (2007), no. 5,494–504.
doi: 10.1016/j.omega.2005.09.002. |
[37] |
E. Zitzler, K. Deb and L. Thiele, Comparison of multiobjective evolutionary algorithms: Empirical results, E Comput., 8 (2000), no. 2,173–195.
doi: 10.1162/106365600568202. |
show all references
References:
[1] |
A. Aguezzoul and P. Ladet, A nonlinear multiobjective approach for the supplier selection, integrating transportation policies, J. Model. Man., 2, (2007), 157–169.
doi: 10.1108/17465660710763434. |
[2] |
N. Aissaoui, M. Haouari and E. Hassini, Supplier selection and order lot sizing modeling: A review, Comput. Oper. Res., 34 (2007), no. 12, 3516–3540.
doi: 10.1016/j.cor.2006.01.016. |
[3] |
S. Amin and G. Zhang, An integrated model for closed-loop supply chain configuration and supplier selection: Multi-objective approach, Expert Syst. Appl., 39 (2012), no. 8, 6782–6791.
doi: 10.1016/j.eswa.2011.12.056. |
[4] |
T. F. Anthony and F. P. Buffa, Strategic purchase scheduling, J. Purch. Mater. Manag., 13 (1977), no. 3, 27–31.
doi: 10.1111/j.1745-493X.1977.tb00400.x. |
[5] |
F. Arikan, A fuzzy solution approach for multi objective supplier selection, Expert Syst. Appl., 40 (2013), no. 3,947–952.
doi: 10.1016/j.eswa.2012.05.051. |
[6] |
E. Babaee Tirkolaee, A. Goli, G.-W. Weber, Multi-Objective Aggregate Production Planning Model Considering Overtime and Outsourcing Options Under Fuzzy Seasonal Demand, in Adv. Manuf. II, Springer, 2019, 81–96. Google Scholar |
[7] |
E. Babaee Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G.-W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, J. of Cleaner Prod., 250 (2020), 119517.
doi: 10.1016/j.jclepro.2019.119517. |
[8] |
M. Bevilacqua, F. E. Ciarapica and G. Giacchetta, A fuzzy-QFD approach to supplier selection, J. Purchase Supp. Man., 12 (2006), no. 1, 14–27.
doi: 10.1016/j.pursup.2006.02.001. |
[9] |
E. Buzdogan-Lindenmayr, G. Kara and A. Selcuk-Kestel, Sevtap Assessment of supplier risk for copper procurement, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (2019), no. 1, 1045–1060.
doi: 10.31801/cfsuasmas.501491. |
[10] |
S. Chou and Y. Chang, A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach, Expert Syst. Appl., 34 (2008), no. 4, 2241–2253.
doi: 10.1016/j.eswa.2007.03.001. |
[11] |
K. Deb and A. Pratap, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Trans. E. Comput., 6 (2002), no. 2,182–197.
doi: 10.1109/4235.996017. |
[12] |
E. A. Demirtas and O. Ustun, Analytic network process and multi-period goal programming integration in purchasing decisions, Comput. Ind. Eng., 56 (2009), no. 2,677–690.
doi: 10.1016/j.cie.2006.12.006. |
[13] |
S. Deng, R. Aydin, C. K. K. Kwong and Y. Huang,
Integrated product line design and supplier selection: A multi-objective optimization paradigm, Comput. Ind. Eng., 70 (2014), 150-158.
doi: 10.1016/j.cie.2014.01.011. |
[14] |
R. M. Ebrahim, J. Razmi and H. Haleh, Scatter search algorithm for supplier selection and order lot sizing under multiple price discount environment, Adv. Eng. Softw., 40 (2009), no. 9,766–776.
doi: 10.1016/j.advengsoft.2009.02.003. |
[15] |
A. Ekici, An improved model for supplier selection under capacity constraint and multiple criteria, Int. J. Prod. Econ., 141 (2013), no. 2,574–581.
doi: 10.1016/j.ijpe.2012.09.013. |
[16] |
R. Farzipoor Sean, A new mathematical approach for suppliers selection: Accounting for non-homogeneity is important, Appl. Math. Comput., 185 (2007), no. 1, 84–95.
doi: 10.1016/j.amc.2006.07.071. |
[17] |
A. Gaballa, Minimum cost allocation of tenders, J. Oper. Res. Soc., 25 (1974), no. 3,389–398.
doi: 10.1057/jors.1974.73. |
[18] |
S. Ghodsypour and C. O$\mathop {\rm{B}}\limits^{'} $rien, The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint, Int. J. Prod. Econ., 73 (2001), no. 1, 15–27.
doi: 10.1016/S0925-5273(01)00093-7. |
[19] |
X. Hu and J. G. Motwani,
Minimizing downside risks for global sourcing under price-sensitive stochastic demand, exchange rate uncertainties and supplier capacity constraints, Int. J. Prod. Econ., 147 (2014), 398-409.
doi: 10.1016/j.ijpe.2013.04.045. |
[20] |
O. Jadidi, S. Zolfaghari and S. Cavalieri,
A new normalized goal programming model for multi-objective problems: A case of supplier selection and order allocation, Int. J. Prod. Econ., 148 (2014), 158-165.
doi: 10.1016/j.ijpe.2013.10.005. |
[21] |
R. Jazemi, J. Gheidar-Kheljani and S. H. Ghodsypour, Modeling the multiobjective problem of supplier selection, taking into account the benefits of the buyer and the suppliers simultaneously, IEEE Trans. Ind. Inform., 7 (2010), no. 3,517–526.
doi: 10.1109/TII.2011.2158835. |
[22] |
D. Kannan, R. Khodaverdi, L. Olfat, A. Jafarian and A. Diabat,
Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain, J. Clean. Prod., 47 (2013), 355-367.
doi: 10.1016/j.jclepro.2013.02.010. |
[23] |
M. Khosroabadi, M. Lotfi and H. Khademizare, Supplier selection program and sizing orders for products at a discounted price and the cost of transportation, J. of Ind. Eng. Res. in Prod. Sys., (2013), no. 1,139–153. Google Scholar |
[24] |
Z. Liao and J. Rittscher, A multi-objective supplier selection model under stochastic demand conditions, 105 (2007), 150–159
doi: 10.1016/j.ijpe.2006.03.001. |
[25] |
R. Lotfi, G.-W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Ind. Manag. Optim., 16 (2020), no. 1,117–140.
doi: 10.3934/jimo.2018143. |
[26] |
G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in multi-objective mathematical programming problems, Appl. Math. Comput., 213 (2009), no. 2,455–465.
doi: 10.1016/j.amc.2009.03.037. |
[27] |
G. Mavrotas and K. Florios, An improved version of the augmented $\varepsilon$-constraint method (AUGMECON 2) for finding the exact pareto set in multi-objective integer programming problems, Appl. Math. Comput., 219 (2013), no. 18, 9652–9669.
doi: 10.1016/j.amc.2013.03.002. |
[28] |
A. Mendoza and J. A. Ventura, Analytical models for supplier selection and order quantity allocation, Appl. Math. Model., 36 (2012), no. 8, 3826–3835.
doi: 10.1016/j.apm.2011.11.025. |
[29] |
K. S. Moghaddam,
Fuzzy multi-objective model for supplier selection and order allocation in reverse logistics systems under supply and demand uncertainty, Expert Syst. Appl., 42 (2015), 6237-6254.
doi: 10.1016/j.eswa.2015.02.010. |
[30] |
D. Mohammaditabar, S. H. Ghodsypour and A. Hafezalkotob,
A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs, Int. J. Prod. Econ., 181 (2016), 87-97.
doi: 10.1016/j.ijpe.2015.11.016. |
[31] |
S. Nazari-Shirkouhi, H. Shakouri, B. Javadi and A. Keramati, Supplier selection and order allocation problem using a two-phase fuzzy multi-objective linear programming, Appl. Math. Model., 37 (2013), no. 22, 9308–9323.
doi: 10.1016/j.apm.2013.04.045. |
[32] |
A. Pan, Allocation of order quantity among suppliers, J. Purch. Mater. Manag., 25 (1989), no. 3, 36–39.
doi: 10.1111/j.1745-493X.1989.tb00489.x. |
[33] |
K. Shaw, R. Shankar, S. S. Yadav and L. S. Thakur, Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain, Expert Syst. Appl., 39 (2012), no. 9, 8182–8192.
doi: 10.1016/j.eswa.2012.01.149. |
[34] |
Y.-C. Tsao and J.-C. Lu, A supply chain network design considering transportation cost discounts, Transp. Res. Part E Logist. Transp. Rev., 48 (2012), no. 2,401–414.
doi: 10.1016/j.tre.2011.10.004. |
[35] |
C. A. Weber, J. R. Current and W. C. Benton, Vendor selection criteria and methods, Eur. J. Oper. Res., 50 (1991), no. 1, 2–18.
doi: 10.1016/0377-2217(91)90033-R. |
[36] |
W. Xia and Z. Wu, Supplier selection with multiple criteria in volume discount environments, Omega, 35 (2007), no. 5,494–504.
doi: 10.1016/j.omega.2005.09.002. |
[37] |
E. Zitzler, K. Deb and L. Thiele, Comparison of multiobjective evolutionary algorithms: Empirical results, E Comput., 8 (2000), no. 2,173–195.
doi: 10.1162/106365600568202. |







Sample No. | No. of suppliers | No. of warehouses | No. of customers | No. of price levels |
1 | 2 | 2 | 3 | 2 |
2 | 2 | 2 | 4 | 2 |
3 | 2 | 3 | 5 | 2 |
4 | 3 | 3 | 4 | 2 |
5 | 3 | 4 | 6 | 2 |
6 | 3 | 5 | 8 | 2 |
7 | 4 | 4 | 8 | 2 |
8 | 4 | 6 | 12 | 2 |
9 | 4 | 8 | 16 | 2 |
10 | 20 | 30 | 40 | 2 |
11 | 25 | 20 | 45 | 2 |
12 | 30 | 25 | 50 | 2 |
13 | 2 | 2 | 4 | 4 |
14 | 2 | 3 | 5 | 4 |
15 | 3 | 3 | 4 | 4 |
16 | 3 | 4 | 6 | 4 |
Sample No. | No. of suppliers | No. of warehouses | No. of customers | No. of price levels |
1 | 2 | 2 | 3 | 2 |
2 | 2 | 2 | 4 | 2 |
3 | 2 | 3 | 5 | 2 |
4 | 3 | 3 | 4 | 2 |
5 | 3 | 4 | 6 | 2 |
6 | 3 | 5 | 8 | 2 |
7 | 4 | 4 | 8 | 2 |
8 | 4 | 6 | 12 | 2 |
9 | 4 | 8 | 16 | 2 |
10 | 20 | 30 | 40 | 2 |
11 | 25 | 20 | 45 | 2 |
12 | 30 | 25 | 50 | 2 |
13 | 2 | 2 | 4 | 4 |
14 | 2 | 3 | 5 | 4 |
15 | 3 | 3 | 4 | 4 |
16 | 3 | 4 | 6 | 4 |
The initial population | 40-30-20 | Mutation rate | 0.1-0.3-0.5 |
Maximum No. of iterations | 400-300-100 | Crossover rate | 0.9-0.7-0.5 |
The initial population | 40-30-20 | Mutation rate | 0.1-0.3-0.5 |
Maximum No. of iterations | 400-300-100 | Crossover rate | 0.9-0.7-0.5 |
The initial population | 30 | Penalty for violation of storage capacity | 30000 |
Maximum No. of iterations | 300 | Penalty for violation of supplier capacity | 30000 |
Crossover rate | 0.7 | Penalty for violation of type Ⅰ carrier capacity | 30000 |
Mutation rate | 0.3 | Penalty for violation of type Ⅱ carrier capacity | 30000 |
The initial population | 30 | Penalty for violation of storage capacity | 30000 |
Maximum No. of iterations | 300 | Penalty for violation of supplier capacity | 30000 |
Crossover rate | 0.7 | Penalty for violation of type Ⅰ carrier capacity | 30000 |
Mutation rate | 0.3 | Penalty for violation of type Ⅱ carrier capacity | 30000 |
Example number | Solutions of the mathematical model | Metaheuristic algorithm | The number of Pareto solutions | Solution time(s) | ||||
Total cost | Lead time | Total cost | Lead time | Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
1 | 68341.34 | 192.79 | 68341.34 | 192.79 | 5 | 23 | 5 | 21 |
38878.15 | 240.99 | 38878.15 | 240.99 | |||||
2 | 121651.55 | 314.90 | 11737.38 | 314.90 | 5 | 24 | 8 | 24 |
64614.87 | 574.81 | 64614.87 | 574.81 | |||||
3 | 75789.91 | 235.26 | 78716.08 | 235.23 | 6 | 27 | 782 | 21 |
53963.32 | 318.05 | 52274.41 | 327.23 | |||||
4 | 78533.08 | 209.05 | 78716.08 | 209.07 | 8 | 27 | 70 | 21 |
43011.82 | 303.94 | 42183.37 | 431.03 | |||||
5 | 81293.42 | 291.36 | 87200.36 | 270.90 | 9 | 38 | 2713 | 22 |
50979.63 | 531.78 | 49854.45 | 645.99 | |||||
6 | 219299.25 | 556.65 | 221539.56 | 591.10 | 5 | 38 | 2471 | 18 |
137964.35 | 1043.08 | 136235.26 | 991.28 | |||||
7 | 150034.73 | 456.26 | 145148.71 | 454.95 | 2 | 28 | 1221 | 17 |
121703.96 | 696.96 | 104719.58 | 738.18 | |||||
8 | - | - | 197548.38 | 454.95 | 0 | 59 | 3600 | 17 |
- | - | 141281.50 | 1336.12 | |||||
9 | - | - | 317966.30 | 1410.04 | 0 | 63 | 7200 | 9 |
- | - | 295530.83 | 1898.55 | |||||
10 | - | - | 692358.54 | 3516.57 | - | 538 | - | 15 |
- | - | 614931.40 | 5127.21 | |||||
11 | - | - | 1395014.66 | 304484.07 | - | 436 | - | 9 |
- | - | 1358331.99 | 305700.94 | |||||
12 | - | - | 1319417.25 | 274612.41 | - | 693 | - | 9 |
- | - | - | - |
Example number | Solutions of the mathematical model | Metaheuristic algorithm | The number of Pareto solutions | Solution time(s) | ||||
Total cost | Lead time | Total cost | Lead time | Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
1 | 68341.34 | 192.79 | 68341.34 | 192.79 | 5 | 23 | 5 | 21 |
38878.15 | 240.99 | 38878.15 | 240.99 | |||||
2 | 121651.55 | 314.90 | 11737.38 | 314.90 | 5 | 24 | 8 | 24 |
64614.87 | 574.81 | 64614.87 | 574.81 | |||||
3 | 75789.91 | 235.26 | 78716.08 | 235.23 | 6 | 27 | 782 | 21 |
53963.32 | 318.05 | 52274.41 | 327.23 | |||||
4 | 78533.08 | 209.05 | 78716.08 | 209.07 | 8 | 27 | 70 | 21 |
43011.82 | 303.94 | 42183.37 | 431.03 | |||||
5 | 81293.42 | 291.36 | 87200.36 | 270.90 | 9 | 38 | 2713 | 22 |
50979.63 | 531.78 | 49854.45 | 645.99 | |||||
6 | 219299.25 | 556.65 | 221539.56 | 591.10 | 5 | 38 | 2471 | 18 |
137964.35 | 1043.08 | 136235.26 | 991.28 | |||||
7 | 150034.73 | 456.26 | 145148.71 | 454.95 | 2 | 28 | 1221 | 17 |
121703.96 | 696.96 | 104719.58 | 738.18 | |||||
8 | - | - | 197548.38 | 454.95 | 0 | 59 | 3600 | 17 |
- | - | 141281.50 | 1336.12 | |||||
9 | - | - | 317966.30 | 1410.04 | 0 | 63 | 7200 | 9 |
- | - | 295530.83 | 1898.55 | |||||
10 | - | - | 692358.54 | 3516.57 | - | 538 | - | 15 |
- | - | 614931.40 | 5127.21 | |||||
11 | - | - | 1395014.66 | 304484.07 | - | 436 | - | 9 |
- | - | 1358331.99 | 305700.94 | |||||
12 | - | - | 1319417.25 | 274612.41 | - | 693 | - | 9 |
- | - | - | - |
Example number | Mathematical model solutions | Metaheuristic algorithm | The number of Pareto solutions | Solution time(s) | ||||
Total cost | Lead time | Total cost | Lead time | Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
13 | 58818.69 | 178.97 | 55591.93 | 178.97 | 12 | 37 | 12 | 16 |
53372.19 | 192.41 | 48951.15 | 438.95 | |||||
14 | 85499.87 | 351.63 | 92925.05 | 92925.05 | 2090 | 40 | 5 | 23 |
62789.30 | 492.64 | 62935.46 | 497.88 | |||||
15 | 83856.96 | 263.83 | 99922.57 | 253.48 | 2113 | 33 | 9 | 22 |
53647.19 | 605.75 | 53384.45 | 593.19 | |||||
16 | 108712.51 | 369.94 | 100694.07 | 394.24 | 2373 | 42 | 6 | 14 |
Example number | Mathematical model solutions | Metaheuristic algorithm | The number of Pareto solutions | Solution time(s) | ||||
Total cost | Lead time | Total cost | Lead time | Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
13 | 58818.69 | 178.97 | 55591.93 | 178.97 | 12 | 37 | 12 | 16 |
53372.19 | 192.41 | 48951.15 | 438.95 | |||||
14 | 85499.87 | 351.63 | 92925.05 | 92925.05 | 2090 | 40 | 5 | 23 |
62789.30 | 492.64 | 62935.46 | 497.88 | |||||
15 | 83856.96 | 263.83 | 99922.57 | 253.48 | 2113 | 33 | 9 | 22 |
53647.19 | 605.75 | 53384.45 | 593.19 | |||||
16 | 108712.51 | 369.94 | 100694.07 | 394.24 | 2373 | 42 | 6 | 14 |
Example No. | Spread criterion | Diversity and uniformity criteria | ||
Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
1 | 29714.36 | 29463.23 | 0.97 | 1.43 |
2 | 54007.81 | 52753.16 | 0.60 | 1.68 |
3 | 19401.56 | 26441.83 | 0.65 | 0.85 |
4 | 14492.95 | 25051.45 | 0.52 | 1.00 |
5 | 25165.95 | 37347.79 | 0.24 | 0.78 |
6 | 41666.65 | 85305.24 | 0.74 | 1.10 |
7 | 28331.79 | 40430.12 | - | 1.00 |
8 | - | 29245.55 | - | 0.93 |
9 | - | 22440.79 | - | 0.74 |
10 | - | 77443.89 | - | 0.92 |
11 | - | 36702.85 | - | 0.80 |
12 | - | 220125.97 | - | 1.51 |
13 | 5446.50 | 6645.87 | 0.68 | 0.99 |
14 | 11106.65 | 2990.05 | 0.32 | 0.77 |
15 | 28204.12 | 46539.36 | 0.24 | 0.97 |
16 | 9979.74 | 15707.09 | 0.72 | 0.93 |
Example No. | Spread criterion | Diversity and uniformity criteria | ||
Mathematical model | Metaheuristic algorithm | Mathematical model | Metaheuristic algorithm | |
1 | 29714.36 | 29463.23 | 0.97 | 1.43 |
2 | 54007.81 | 52753.16 | 0.60 | 1.68 |
3 | 19401.56 | 26441.83 | 0.65 | 0.85 |
4 | 14492.95 | 25051.45 | 0.52 | 1.00 |
5 | 25165.95 | 37347.79 | 0.24 | 0.78 |
6 | 41666.65 | 85305.24 | 0.74 | 1.10 |
7 | 28331.79 | 40430.12 | - | 1.00 |
8 | - | 29245.55 | - | 0.93 |
9 | - | 22440.79 | - | 0.74 |
10 | - | 77443.89 | - | 0.92 |
11 | - | 36702.85 | - | 0.80 |
12 | - | 220125.97 | - | 1.51 |
13 | 5446.50 | 6645.87 | 0.68 | 0.99 |
14 | 11106.65 | 2990.05 | 0.32 | 0.77 |
15 | 28204.12 | 46539.36 | 0.24 | 0.97 |
16 | 9979.74 | 15707.09 | 0.72 | 0.93 |
Methodology | Average processing time (in seconds) | The number of Pareto solutions |
augmented epsilon-constraint method | 1897 | 6 |
Metaheuristic algorithm | 38 | 19 |
Methodology | Average processing time (in seconds) | The number of Pareto solutions |
augmented epsilon-constraint method | 1897 | 6 |
Metaheuristic algorithm | 38 | 19 |
[1] |
Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021037 |
[2] |
Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023 |
[3] |
Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169 |
[4] |
Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021024 |
[5] |
Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021035 |
[6] |
Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 |
[7] |
Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 |
[8] |
Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 |
[9] |
Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021008 |
[10] |
Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 |
[11] |
Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020136 |
[12] |
Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020134 |
[13] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[14] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[15] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021019 |
[16] |
Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 |
[17] |
Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 |
[18] |
Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 |
[19] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[20] |
Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]