Document Type : Article

**Authors**

Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran

**Abstract**

Selecting the most suitable optimal point among the Pareto optimal points could help the experts to make an appropriate decision in an uncertain and complex situation. In this paper, an evaluating and ranking approach is proposed based on hesitant fuzzy set environment to assess the obtained Pareto optimal points from the proposed bi-objective multi-echelon supply chain model with locating distribution centers. In this respect, the proposed model has elaborated for perishable products based on fuzzy customers' demand. To address the issue, the possibilistic chance constraint programming approach has manipulated based on the trapezoidal fuzzy membership function. Moreover, the proposed hesitant fuzzy ranking approach is constructed based on group decision analysis and the last aggregation approach. Thereby, the last aggregation approach by aggregating the experts' opinions in last step could prevent the data loss. However, a case study about the perishable dairy products is considered to indicate the applicability of the proposed bi-objective multi-echelon supply chain model with locating distribution centers. Finally, a comparative analysis is provided between the obtained results and the current practice to show the feasibility and efficiency of the proposed model.

**Keywords**

- Multi-echelon supply chain, Pareto optimal solution
- Perishable products
- Group decision analysis
- Possibilistic approach

**Main Subjects**

References:

1. Ebrahimnejad, S., Mousavi, S.M., and H. Seyrafianpour "Risk identification and assessment for buildoperate- transfer projects: A fuzzy multi attribute decision making model", Expert Systems with Applications, 37(1), pp. 575-586 (2010).

2. Vahdani, B., Mousavi, S.M., Hashemi, H., Mousakhani, M., and Tavakkoli-Moghaddam, R. "A new compromise solution method for fuzzy group decision-making problems with an application to the contractor selection", Engineering Applications of Artificial Intelligence, 26(2), pp. 779-788 (2013).

3. Ze-shui, X. "A multi-attribute group decision making method based on term indices in linguistic evaluation scales [J]", Journal of Systems Engineering, 1, p. 13 (2005).

4. Makui, A., Mojtahedi, S.M., and Mousavi, S.M. "Project risk identification and analysis based on group decision making methodology in a fuzzy environment", International Journal of Management Science and Engineering Management, 5(2), pp. 108-118 (2010).

5. Zadeh, L.A. "Fuzzy sets", Information and Control, 8(3), pp. 338-353 (1965).

6. Dubois, D.J., Fuzzy Sets and Systems: Theory and Applications, 144, Academic Press (1980).

7. Miyamoto, S., Multisets and Fuzzy Multisets, in Soft Computing and Human-Centered Machines, Springer, pp. 9-33 (2000).

8. Turksen, I.B. "Interval valued fuzzy sets based on normal forms", Fuzzy Sets and Systems, 20(2), pp. 191-210 (1986).

9. Zadeh, L.A. "The concept of a linguistic variable and its application to approximate reasoning-I", Information Sciences, 8(3), pp. 199-249 (1975).

10. Atanassov, K.T. "Intuitionistic fuzzy sets", Fuzzy sets and Systems, 20(1), pp. 87-96 (1986).

11. Torra, V. and Narukawa, Y. "On hesitant fuzzy sets and decision", In Fuzzy Systems", FUZZ-IEEE 2009. IEEE International Conference on, IEEE (2009).

12. Torra, V. "Hesitant fuzzy sets", International Journal of Intelligent Systems, 25(6), pp. 529-539 (2010).

13. Zhang, N. and Wei, G. "Extension of VIKOR method for decision making problem based on hesitant fuzzy set", Applied Mathematical Modelling, 37(7), pp. 4938- 4947 (2013).

14. Xu, Z. and Zhang, X. "Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information", Knowledge-Based Systems, 52, pp. 53-64 (2013).

15. Wei, G. and Zhang, N. "A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method", Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 26(2), pp. 1065-1075 (2014).

16. Chen, N. and Xu, Z. "Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems", Information Sciences, 292, pp. 175-197 (2015).

17. Joshi, D. and Kumar, S. "Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making", European Journal of Operational Research, 248(1), pp. 183-191 (2016).

18. Qin, J., Liu, X., and Pedrycz, W. "An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment", European Journal of Operational Research, 258(2), pp. 626-638 (2017).

19. Ghodratnama, A., Tavakkoli-Moghaddam, R., and Azaron, A. "A fuzzy possibilistic bi-objective hub covering problem considering production facilities, time horizons and transporter vehicles", The International Journal of Advanced Manufacturing Technology, 66(1- 4), pp. 187-206 (2013).

20. Mohammadi, M., Jolai, F., and Tavakkoli- Moghaddam, R. "Solving a new stochastic multi-mode p-hub covering location problem considering risk by a novel multi-objective algorithm", Applied Mathematical Modelling, 37(24), pp. 10053-10073

(2013).

21. Rahimi, M., Baboli, A., and Rekik, Y. "A bi-objective inventory routing problem by considering customer satisfaction level in context of perishable product", In Computational Intelligence in Production and Logistics Systems (CIPLS), 2014 IEEE Symposium on, IEEE (2014).

22. Ebrahimi Zade, A., Sadegheih, A., and Lotfi, M.M. "A modified NSGA-II solution for a new multi-objective hub maximal covering problem under uncertain shipments", Journal of Industrial Engineering International, 10(4), pp. 185-197 (2014).

23. Pasandideh, S.H.R., Niaki, S.T.A., and Asadi, K. Biobjective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-II and NRGA", Information Sciences, 292, pp. 57-74 (2015).

24. Pasandideh, S.H.R., Niaki, S.T.A., and Asadi, K. "Optimizing a bi-objective multi-product multi-period three echelon supply chain network with warehouse reliability", Expert Systems with Applications, 42(5), pp. 2615-2623 (2015).

25. Khalili-Damghani, K., Abtahi, A.-R., and Ghasemi, A. "A new bi-objective location-routing problem for distribution of perishable products: Evolutionary computation approach", Journal of Mathematical Modelling and Algorithms in Operations Research, 14(3), pp. 287-312 (2015).

26. Sarrafha, K., Rahmati, S.H.A., Niaki, S.T.A., and Zaretalab, A. "A bi-objective integrated procurement, production, and distribution problem of a multiechelon supply chain network design: A new tuned MOEA", Computers & Operations Research, 54, pp. 35-51 (2015).

27. Alavidoost, M., Tarimoradi, M., and Zarandi, M.F. "Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks", Journal of Intelligent Manufacturing, 29(4), pp. 1-18 (2015).

28. Ghodratnama, A., Tavakkoli-Moghaddam, R., and Azaron, A. "Robust and fuzzy goal programming optimization approaches for a novel multi-objective hub location-allocation problem: A supply chain overview", Applied Soft Computing, 37, pp. 255-276 (2015).

29. Pasandideh, S.H.R., Niaki, S.T.A., and Sheikhi, M. "A bi-objective hub maximal covering location problem considering time-dependent reliability and the second type of coverage", International Journal of Management Science and Engineering Management, 11(4), pp. 1-8 (2015).

30. Maghsoudlou, H., Kahag, M.R., Niaki, S.T.A., and Pourvaziri, H. "Bi-objective optimization of a threeechelon multi-server supply-chain problem in congested systems: Modeling and solution", Computers & Industrial Engineering, 99, pp. 41-62 (2016).

31. Ghezavati, V. and Beigi, M. "Solving a bi-objective mathematical model for location-routing problem with time windows in multi-echelon reverse logistics using metaheuristic procedure", Journal of Industrial Engineering International, 12(4), pp. 469-483 (2016).

32. Ebrahimi, S.B. "A bi-objective model for a multiechelon supply chain design considering efficiency and customer satisfaction: a case study in plastic parts industry", The International Journal of Advanced Manufacturing Technology, 95(9-12), pp. 1-19 (2017).

33. Habibi-Kouchaksaraei, M., Paydar, M.M., and Asadi-Gangraj, E. "Designing a bi-objective multi-echelon robust blood supply chain in a disaster", Applied Mathematical Modelling, 55, pp. 583-599 (2018).

34. Farahani, P., Grunow, M., and Gunther, H.O. "Integrated production and distribution planning for perishable food products", Flexible Services and Manufacturing Journal, 24(1), pp. 28-51 (2012).

35. Xia, M. and Xu, Z. "Hesitant fuzzy information aggregation in decision making", International Journal of Approximate Reasoning, 52(3), pp. 395-407 (2011).

36. Yue, Z. "An extended TOPSIS for determining weights of decision makers with interval numbers", Knowledge- Based Systems, 24(1), pp. 146-153 (2011).

37. Zhu, B., Xu, Z., and Xia, M. "Hesitant fuzzy geometric Bonferroni means", Information Sciences, 205, pp. 72- 85 (2012).

38. Liu, B. and Iwamura, K. "Chance constrained programming with fuzzy parameters", Fuzzy Sets and Systems, 94(2), pp. 227-237 (1998).

39. Inuiguchi, M., Ichihashi, H., and Kume, Y. "Modality constrained programming problems: a unified approach to fuzzy mathematical programming problems in the setting of possibility theory", Information Sciences, 67(1), pp. 93-126 (1993).

40. Pishvaee, M., Razmi, J., and Torabi, S.A. "Robust possibilistic programming for socially responsible supply chain network design: A new approach", Fuzzy Sets and Systems, 206, pp. 1-20 (2012).

41. Dubois, D. and Prade, H. "The mean value of a fuzzy number", Fuzzy Sets and Systems, 24(3), pp. 279-300 (1987).

42. Heilpern, S. "The expected value of a fuzzy number", Fuzzy Sets and Systems, 47(1), pp. 81-86 (1992).

43. Pishvaee, M.S., Rabbani, M., and Torabi, S.A. "A robust optimization approach to closed-loop supply chain network design under uncertainty", Applied Mathematical Modelling, 35(2), pp. 637-649 (2011).

44. Zhang, W. and Reimann, M. "A simple augmented constraint method for multi-objective mathematical integer programming problems", European Journal of Operational Research, 234(1), pp. 15-24 (2014).

45. Mavrotas, G. "Effective implementation of the "- constraint method in multi-objective mathematical programming problems", Applied Mathematics and Computation, 213(2), pp. 455-465 (2009).

2. Vahdani, B., Mousavi, S.M., Hashemi, H., Mousakhani, M., and Tavakkoli-Moghaddam, R. "A new compromise solution method for fuzzy group decision-making problems with an application to the contractor selection", Engineering Applications of Artificial Intelligence, 26(2), pp. 779-788 (2013).

3. Ze-shui, X. "A multi-attribute group decision making method based on term indices in linguistic evaluation scales [J]", Journal of Systems Engineering, 1, p. 13 (2005).

4. Makui, A., Mojtahedi, S.M., and Mousavi, S.M. "Project risk identification and analysis based on group decision making methodology in a fuzzy environment", International Journal of Management Science and Engineering Management, 5(2), pp. 108-118 (2010).

5. Zadeh, L.A. "Fuzzy sets", Information and Control, 8(3), pp. 338-353 (1965).

6. Dubois, D.J., Fuzzy Sets and Systems: Theory and Applications, 144, Academic Press (1980).

7. Miyamoto, S., Multisets and Fuzzy Multisets, in Soft Computing and Human-Centered Machines, Springer, pp. 9-33 (2000).

8. Turksen, I.B. "Interval valued fuzzy sets based on normal forms", Fuzzy Sets and Systems, 20(2), pp. 191-210 (1986).

9. Zadeh, L.A. "The concept of a linguistic variable and its application to approximate reasoning-I", Information Sciences, 8(3), pp. 199-249 (1975).

10. Atanassov, K.T. "Intuitionistic fuzzy sets", Fuzzy sets and Systems, 20(1), pp. 87-96 (1986).

11. Torra, V. and Narukawa, Y. "On hesitant fuzzy sets and decision", In Fuzzy Systems", FUZZ-IEEE 2009. IEEE International Conference on, IEEE (2009).

12. Torra, V. "Hesitant fuzzy sets", International Journal of Intelligent Systems, 25(6), pp. 529-539 (2010).

13. Zhang, N. and Wei, G. "Extension of VIKOR method for decision making problem based on hesitant fuzzy set", Applied Mathematical Modelling, 37(7), pp. 4938- 4947 (2013).

14. Xu, Z. and Zhang, X. "Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information", Knowledge-Based Systems, 52, pp. 53-64 (2013).

15. Wei, G. and Zhang, N. "A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method", Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 26(2), pp. 1065-1075 (2014).

16. Chen, N. and Xu, Z. "Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems", Information Sciences, 292, pp. 175-197 (2015).

17. Joshi, D. and Kumar, S. "Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making", European Journal of Operational Research, 248(1), pp. 183-191 (2016).

18. Qin, J., Liu, X., and Pedrycz, W. "An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment", European Journal of Operational Research, 258(2), pp. 626-638 (2017).

19. Ghodratnama, A., Tavakkoli-Moghaddam, R., and Azaron, A. "A fuzzy possibilistic bi-objective hub covering problem considering production facilities, time horizons and transporter vehicles", The International Journal of Advanced Manufacturing Technology, 66(1- 4), pp. 187-206 (2013).

20. Mohammadi, M., Jolai, F., and Tavakkoli- Moghaddam, R. "Solving a new stochastic multi-mode p-hub covering location problem considering risk by a novel multi-objective algorithm", Applied Mathematical Modelling, 37(24), pp. 10053-10073

(2013).

21. Rahimi, M., Baboli, A., and Rekik, Y. "A bi-objective inventory routing problem by considering customer satisfaction level in context of perishable product", In Computational Intelligence in Production and Logistics Systems (CIPLS), 2014 IEEE Symposium on, IEEE (2014).

22. Ebrahimi Zade, A., Sadegheih, A., and Lotfi, M.M. "A modified NSGA-II solution for a new multi-objective hub maximal covering problem under uncertain shipments", Journal of Industrial Engineering International, 10(4), pp. 185-197 (2014).

23. Pasandideh, S.H.R., Niaki, S.T.A., and Asadi, K. Biobjective optimization of a multi-product multi-period three-echelon supply chain problem under uncertain environments: NSGA-II and NRGA", Information Sciences, 292, pp. 57-74 (2015).

24. Pasandideh, S.H.R., Niaki, S.T.A., and Asadi, K. "Optimizing a bi-objective multi-product multi-period three echelon supply chain network with warehouse reliability", Expert Systems with Applications, 42(5), pp. 2615-2623 (2015).

25. Khalili-Damghani, K., Abtahi, A.-R., and Ghasemi, A. "A new bi-objective location-routing problem for distribution of perishable products: Evolutionary computation approach", Journal of Mathematical Modelling and Algorithms in Operations Research, 14(3), pp. 287-312 (2015).

26. Sarrafha, K., Rahmati, S.H.A., Niaki, S.T.A., and Zaretalab, A. "A bi-objective integrated procurement, production, and distribution problem of a multiechelon supply chain network design: A new tuned MOEA", Computers & Operations Research, 54, pp. 35-51 (2015).

27. Alavidoost, M., Tarimoradi, M., and Zarandi, M.F. "Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks", Journal of Intelligent Manufacturing, 29(4), pp. 1-18 (2015).

28. Ghodratnama, A., Tavakkoli-Moghaddam, R., and Azaron, A. "Robust and fuzzy goal programming optimization approaches for a novel multi-objective hub location-allocation problem: A supply chain overview", Applied Soft Computing, 37, pp. 255-276 (2015).

29. Pasandideh, S.H.R., Niaki, S.T.A., and Sheikhi, M. "A bi-objective hub maximal covering location problem considering time-dependent reliability and the second type of coverage", International Journal of Management Science and Engineering Management, 11(4), pp. 1-8 (2015).

30. Maghsoudlou, H., Kahag, M.R., Niaki, S.T.A., and Pourvaziri, H. "Bi-objective optimization of a threeechelon multi-server supply-chain problem in congested systems: Modeling and solution", Computers & Industrial Engineering, 99, pp. 41-62 (2016).

31. Ghezavati, V. and Beigi, M. "Solving a bi-objective mathematical model for location-routing problem with time windows in multi-echelon reverse logistics using metaheuristic procedure", Journal of Industrial Engineering International, 12(4), pp. 469-483 (2016).

32. Ebrahimi, S.B. "A bi-objective model for a multiechelon supply chain design considering efficiency and customer satisfaction: a case study in plastic parts industry", The International Journal of Advanced Manufacturing Technology, 95(9-12), pp. 1-19 (2017).

33. Habibi-Kouchaksaraei, M., Paydar, M.M., and Asadi-Gangraj, E. "Designing a bi-objective multi-echelon robust blood supply chain in a disaster", Applied Mathematical Modelling, 55, pp. 583-599 (2018).

34. Farahani, P., Grunow, M., and Gunther, H.O. "Integrated production and distribution planning for perishable food products", Flexible Services and Manufacturing Journal, 24(1), pp. 28-51 (2012).

35. Xia, M. and Xu, Z. "Hesitant fuzzy information aggregation in decision making", International Journal of Approximate Reasoning, 52(3), pp. 395-407 (2011).

36. Yue, Z. "An extended TOPSIS for determining weights of decision makers with interval numbers", Knowledge- Based Systems, 24(1), pp. 146-153 (2011).

37. Zhu, B., Xu, Z., and Xia, M. "Hesitant fuzzy geometric Bonferroni means", Information Sciences, 205, pp. 72- 85 (2012).

38. Liu, B. and Iwamura, K. "Chance constrained programming with fuzzy parameters", Fuzzy Sets and Systems, 94(2), pp. 227-237 (1998).

39. Inuiguchi, M., Ichihashi, H., and Kume, Y. "Modality constrained programming problems: a unified approach to fuzzy mathematical programming problems in the setting of possibility theory", Information Sciences, 67(1), pp. 93-126 (1993).

40. Pishvaee, M., Razmi, J., and Torabi, S.A. "Robust possibilistic programming for socially responsible supply chain network design: A new approach", Fuzzy Sets and Systems, 206, pp. 1-20 (2012).

41. Dubois, D. and Prade, H. "The mean value of a fuzzy number", Fuzzy Sets and Systems, 24(3), pp. 279-300 (1987).

42. Heilpern, S. "The expected value of a fuzzy number", Fuzzy Sets and Systems, 47(1), pp. 81-86 (1992).

43. Pishvaee, M.S., Rabbani, M., and Torabi, S.A. "A robust optimization approach to closed-loop supply chain network design under uncertainty", Applied Mathematical Modelling, 35(2), pp. 637-649 (2011).

44. Zhang, W. and Reimann, M. "A simple augmented constraint method for multi-objective mathematical integer programming problems", European Journal of Operational Research, 234(1), pp. 15-24 (2014).

45. Mavrotas, G. "Effective implementation of the "- constraint method in multi-objective mathematical programming problems", Applied Mathematics and Computation, 213(2), pp. 455-465 (2009).

Transactions on Industrial Engineering (E)

September and October 2019Pages 2952-2970