Figure 1.
Demonstration of a chromosome (i.e., a solution) in the proposed metaheuristic algorithm
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Equilibrium strategies in a supply chain with capital constrained suppliers: The impact of external financing
doi: 10.3934/jimo.2020134
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.
Keywords:
Three-level supply chain,
supplier selection,
shipping discounts,
multi-objective optimization,
NSGA-Ⅱ.
Mathematics Subject Classification:
Primary: 90B06.
Citation:
Alireza Eydi, Rozhin Saedi.
A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020134
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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. |
Figure 2.
Demonstration of the solution of the case study
Figure 4.
Demonstration of the mutation operator
Figure 4.
Demonstration of the mutation operator
Figure 5.
Setting the parameters of the algorithm
Figure 6.
Comparison of the Pareto front for a numerical example
Figure 7.
The Pareto front for a large-scale example
Table 1.
Details of the sample problems
| 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 |
Table 2.
Parametrization of the algorithm
| 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 |
Table 3.
Results of the parametrization of the proposed metaheuristic algorithm
| 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 |
Table 4.
Sample problems with two price levels
| 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 |
| - | - | - | - | |||||
Table 5.
Sample problems with four price levels
| 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 |
Table 6.
Comparison of the proposed mathematical model and algorithm based on diversity and spread criteria
| 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 |
Table 7.
Comparison between different solution methods regarding average processing time and the number of Pareto solutions obtained for the small-sized problem
| 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 |
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