# American Institute of Mathematical Sciences

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

* Corresponding author: Alireza Eydi

Received  December 2019 Revised  June 2020 Published  August 2020

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.

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
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. M. Ebrahim, J. Razmi and H. 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Econ., 73 (2001), no. 1, 15–27. doi: 10.1016/S0925-5273(01)00093-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] A. Gaballa, Minimum cost allocation of tenders, J. Oper. Res. Soc., 25 (1974), no. 3,389–398. doi: 10.1057/jors.1974.73.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Demonstration of a chromosome (i.e., a solution) in the proposed metaheuristic algorithm
Demonstration of the solution of the case study
Demonstration of the mutation operator
Demonstration of the mutation operator
Setting the parameters of the algorithm
Comparison of the Pareto front for a numerical example
The Pareto front for a large-scale example
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
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
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
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 - - - -
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
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
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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