A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints

Alireza Eydi; Rozhin Saedi

doi:10.3934/jimo.2020134

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doi: 10.3934/jimo.2020134

A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints

Alireza Eydi ^1,, and Rozhin Saedi ^2,

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.

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

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

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

Figure 1. Demonstration of a chromosome (i.e., a solution) in the proposed metaheuristic algorithm

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Figure 2. Demonstration of the solution of the case study

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Figure 4. Demonstration of the mutation operator

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Figure 4. Demonstration of the mutation operator

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Figure 5. Setting the parameters of the algorithm

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Figure 6. Comparison of the Pareto front for a numerical example

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Figure 7. The Pareto front for a large-scale example

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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

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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

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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

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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)
Example number	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
1	38878.15	240.99	38878.15	240.99	5	23	5	21
2	121651.55	314.90	11737.38	314.90	5	24	8	24
2	64614.87	574.81	64614.87	574.81	5	24	8	24
3	75789.91	235.26	78716.08	235.23	6	27	782	21
3	53963.32	318.05	52274.41	327.23	6	27	782	21
4	78533.08	209.05	78716.08	209.07	8	27	70	21
4	43011.82	303.94	42183.37	431.03	8	27	70	21
5	81293.42	291.36	87200.36	270.90	9	38	2713	22
5	50979.63	531.78	49854.45	645.99	9	38	2713	22
6	219299.25	556.65	221539.56	591.10	5	38	2471	18
6	137964.35	1043.08	136235.26	991.28	5	38	2471	18
7	150034.73	456.26	145148.71	454.95	2	28	1221	17
7	121703.96	696.96	104719.58	738.18	2	28	1221	17
8	-	-	197548.38	454.95	0	59	3600	17
8	-	-	141281.50	1336.12	0	59	3600	17
9	-	-	317966.30	1410.04	0	63	7200	9
9	-	-	295530.83	1898.55	0	63	7200	9
10	-	-	692358.54	3516.57	-	538	-	15
10	-	-	614931.40	5127.21	-	538	-	15
11	-	-	1395014.66	304484.07	-	436	-	9
11	-	-	1358331.99	305700.94	-	436	-	9
12	-	-	1319417.25	274612.41	-	693	-	9
12	-	-	-	-	-	693	-	9

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Example number	Solutions of the mathematical model		Metaheuristic algorithm		The number of Pareto solutions		Solution time(s)
Example number	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
1	38878.15	240.99	38878.15	240.99	5	23	5	21
2	121651.55	314.90	11737.38	314.90	5	24	8	24
2	64614.87	574.81	64614.87	574.81	5	24	8	24
3	75789.91	235.26	78716.08	235.23	6	27	782	21
3	53963.32	318.05	52274.41	327.23	6	27	782	21
4	78533.08	209.05	78716.08	209.07	8	27	70	21
4	43011.82	303.94	42183.37	431.03	8	27	70	21
5	81293.42	291.36	87200.36	270.90	9	38	2713	22
5	50979.63	531.78	49854.45	645.99	9	38	2713	22
6	219299.25	556.65	221539.56	591.10	5	38	2471	18
6	137964.35	1043.08	136235.26	991.28	5	38	2471	18
7	150034.73	456.26	145148.71	454.95	2	28	1221	17
7	121703.96	696.96	104719.58	738.18	2	28	1221	17
8	-	-	197548.38	454.95	0	59	3600	17
8	-	-	141281.50	1336.12	0	59	3600	17
9	-	-	317966.30	1410.04	0	63	7200	9
9	-	-	295530.83	1898.55	0	63	7200	9
10	-	-	692358.54	3516.57	-	538	-	15
10	-	-	614931.40	5127.21	-	538	-	15
11	-	-	1395014.66	304484.07	-	436	-	9
11	-	-	1358331.99	305700.94	-	436	-	9
12	-	-	1319417.25	274612.41	-	693	-	9
12	-	-	-	-	-	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)
Example number	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
13	53372.19	192.41	48951.15	438.95	12	37	12	16
14	85499.87	351.63	92925.05	92925.05	2090	40	5	23
14	62789.30	492.64	62935.46	497.88	2090	40	5	23
15	83856.96	263.83	99922.57	253.48	2113	33	9	22
15	53647.19	605.75	53384.45	593.19	2113	33	9	22
16	108712.51	369.94	100694.07	394.24	2373	42	6	14
16					2373	42	6	14

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Example number	Mathematical model solutions		Metaheuristic algorithm		The number of Pareto solutions		Solution time(s)
Example number	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
13	53372.19	192.41	48951.15	438.95	12	37	12	16
14	85499.87	351.63	92925.05	92925.05	2090	40	5	23
14	62789.30	492.64	62935.46	497.88	2090	40	5	23
15	83856.96	263.83	99922.57	253.48	2113	33	9	22
15	53647.19	605.75	53384.45	593.19	2113	33	9	22
16	108712.51	369.94	100694.07	394.24	2373	42	6	14
16					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
Example No.	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

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Example No.	Spread criterion		Diversity and uniformity criteria
Example No.	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

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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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American Institute of Mathematical Sciences

A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints

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