An improved MULTIMOORA method for multi-valued neutrosophic multi-criteria group decision-making based on prospect theory

Document Type : Article


1 a- School of Business, Central South University, Changsha 410083, PR China.

2 - College of Tourism, Hunan Normal University, Changsha 410081, PR China. - College of Logistics and Transportation, Central South University of Forestry and Technology, Changsha 410004, PR China.


With the development of the economy and society, the scale of cities is increasing. At the same time, there are many subways being constructed in many cities. In the construction of subways, an appropriate scheme plays an important role in reducing cost and improving the satisfaction of the public. This paper attaches great importance to present a multi-criteria group decision-making (MCGDM) method to deal with selecting an appropriate construction scheme for subways. The process of selecting an appropriate construction scheme for subways is complex because it includes a great deal of fuzzy and uncertain information which can be presented by multi-valued neutrosophic numbers (MVNNs). In addition, in order to handle the interaction of inputs, an improved generalized multi-valued neutrosophic weighted Heronian mean (IGMVNWHM) operator is introduced. Subsequently, a new distance measure between two MVNNs is defined for deriving the objective criteria weights. Furthermore, considering the bounded rationality of decision-makers, we develop an improved multi-valued neutrosophic MULTIMOORA method based on prospect theory (IMVN-PT-MULTIMOORA). Finally, an application example of selecting an appropriate construction scheme for a subway and the influence of parameter are described. In addition, the proposed approach is compared with some existing methods to prove its validity and advantages.


1. Chen, Y.C., Hsu, L.H., and Tan, J.J.M. "A recursively construction scheme for super fault-tolerant hamiltonian graphs", Applied Mathematics & Computation, 177(2), pp. 465-481 (2006).
2. Wang, Y., Wang, J.Q., and Wang, T.L. "Fuzzy stochastic multi-criteria decision-making methods with interval neutrosophic probability based on regret theory", Journal of Intelligent & Fuzzy Systems, 35(2), pp. 2309-2322 (2018).
3. Ji, P., Wang, J.Q., and Zhang, H.Y. "Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers", Neural Computing & Applications, 30(3), pp. 1-25 (2018).
4. Zadeh, L.A. "Fuzzy sets", Information & Control, 8(3), pp. 338-353 (1965).
5. Luo, S.Z., Zhang, H.Y., Wang, J.Q., et al. "Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations", Journal of the Operational Research Society, 70(12), pp. 2039-2055 (2018).
6. Nie, R.X., Tian, Z.P., Wang, J.Q., et al. "Pythagorean fuzzy multiple criteria decision analysis based on Shapley fuzzy measures and partitioned normalized weighted Bonferroni mean operator", International Journal of Intelligent Systems, 34(2), pp. 297-324 (2018).
7. Shao, X., Si, H., and Zhang, W. "Fuzzy wavelet neural control with improved prescribed performance for MEMS gyroscope subject to input quantization", Fuzzy Sets and Systems, 411, pp. 136-154 (2021).
8. Wang, X.K., Zhang, H.Y., Wang, J.Q., et al. "Extended Todim-Promenthee II method with method with hesitant probabilistic information for solving potential risk evaluation problems of water resource carrying capacity", Expert Systems, e12681 (2021).
9. Wang, X.K., Wang, S.H., Zhang, H.Y., et al. "The recommendation method for hotel selection under traveller preference characteristics: A cloud-based multicriteria group decision support model", Group Decision and Negotiation, 30(6), pp. 1433-1469 (2021).
10. Atanassov, K.T. "Intuitionistic fuzzy sets", Fuzzy Sets & Systems, 20(1), pp. 87-96 (1986).
11. Atanassov, K.T. "Two theorems for intuitionistic fuzzy sets", Fuzzy Sets & Systems, 110(2), pp. 267-269 (2000).
12. Atanassov, K.T. "New operations defined over the intuitionistic fuzzy sets", Fuzzy Sets & Systems, 61(2), pp. 137-142 (1994).
13. Ghosh, S.K., Biswas, B., and Ghosh, A. "Development of intuitionistic fuzzy special embedded convolutional neural network for mammography enhancement", Computational Intelligence, 37(1), pp. 47-69 (2021).
14. Kuo, R.J., Cheng, W.C., Lien, W.-C., et al. "Application of genetic algorithm-based intuitionistic fuzzy neural network to medical cost forecasting for acute hepatitis patients in emergency room", Journal of Intelligent & Fuzzy Systems, 37(4), pp. 5455-5469 (2019).
15. Hu, J.H., Pan, L., Yang, Y., et al. "A group medical diagnosis model based on intuitionistic fuzzy soft sets", Applied Soft Computing, 77, pp. 453-466 (2019).
16. Atanassov, K. "Interval valued intuitionistic fuzzy sets", Fuzzy Sets & Systems, 31(3), pp. 343-349 (1989).
17. Li, J. and Wang, J.Q. "Multi-criteria decision making with probabilistic hesitant fuzzy information based on expected  multiplicative consistency", Neural Computing and Applications, 31(12), pp. 8897-8915 (2018).
18. Hu, J.H., Yang, Y., Zhang, X.L., et al. "Similarity and entropy measures for hesitant fuzzy sets", International Transactions in Operational Research, 25(3), pp. 857-886 (2018).
19. Mishra, A.R., Rani, P., Krishankumar, R., et al. "An extended fuzzy decision-making framework using hesitant fuzzy sets for the drug selection to treat the mild symptoms of Coronavirus Disease 2019 (COVID- 19)", Applied Soft Computing, 103, 107155 (2021).
20. Zhu, B., Xu, Z., and Xia, M. "Dual hesitant fuzzy sets", Journal of Applied Mathematics, 11(3), pp. 1- 13 (2012).
21. Qian, G., Wang, H., and Feng, X. "Generalized hesitant fuzzy sets and their application in decision support system", Knowledge-Based Systems, 37(4), pp. 357-365 (2013).
22. Wang, H., Smarandache, F., Zhang, Y., et al. "Single valued neutrosophic sets", Review of the Air Force Academy, 10 (2010).
23. Smarandache, F. "A unifying field in logics: Neutrosophic logic", Multiple-Valued Logic, 8(3), pp. 489-503 (1999).
24. Smarandache, F. "Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), pythagorean fuzzy set (atanassov's intuitionistic fuzzy set of second type), q-rung orthopair fuzzy set, spherical fuzzy set, and n-HyperSpherical fuzzy set, while neutrosophication is a generalization of regret
theory, grey system theory, and three-ways decision (revisited)", Journal of New Theory, (29), pp. 1-31 (2019).
25. Rivieccio, U. "Neutrosophic logics: Prospects and problems", Fuzzy Sets & Systems, 159(14), pp. 1860- 1868 (2008).
26. Broumi, S., Bakali, A., Talea, M., et al. "Isolated single valued neutrosophic graphs", Neutrosophic Sets & Systems, 11, pp. 74-78 (2016).
27. Sahin, R. and Kucuk, A. "Subsethood measure for single valued neutrosophic sets", Journal of Intelligent & Fuzzy Systems, 29(2), pp. 525-530 (2015).
28. Li, Y.Y., Wang, J.Q., and Wang, T.L. "A linguistic neutrosophic multi-criteria group decision-making approach with EDAS method", Arabian Journal for Science and Engineering, 44(3), pp. 2737-2749 (2018).
29. Liang, R.X., Wang, J.Q., and Zhang, H.Y. "A multi-criteria decision-making method based on singlevalued trapezoidal neutrosophic preference relations with complete weight information", Neural Computing and Applications, 30(11), pp. 3383-3398 (2017).
30. Tian, Z.P., Wang, J., Wang, J.Q., et al. "Simplified neutrosophic linguistic multi-criteria group decisionmaking approach to green product development", Group Decision & Negotiation, 26(3), pp. 597-627 (2017).
31. Liu, P., Zhang, L., Liu, X., et al. "Multi-valued neutrosophic number bonferroni mean operators with their applications in multiple attribute group decision making", International Journal of Information Technology & Decision Making, 15(5), pp. 1181-1210 (2016).
32. Peng, J.J., Wang, J.Q., and Yang, W.E. "A multi-valued neutrosophic qualitative  flexible approach based on likelihood for multi-criteria decision-making problems", International Journal of Systems Science, 48(2), pp. 425-435 (2017).
33. Peng, J., Tian, C., Zhang, W., et al. "An integrated multi-criteria decision-making framework for sustainable supplier selection under picture fuzzy environment", Technological and Economic Development of Economy, 26(3), pp. 573-598 (2020).
34. Wang, J., Wang, J.Q., Tian, Z.P., et al. "A multihesitant fuzzy linguistic multicriteria decision-making approach for logistics outsourcing with incomplete weight information", International Transactions in Operational Research, 25(9), pp. 831-856 (2017).
35. Song, M., Jiang, W., Xie, C., et al. "A new interval numbers power average operator in multiple attribute decision making", International Journal of Intelligent Systems, 32(6), pp. 631-644 (2016).
36. Li, Y. andWang, Q.S. "Research on hotel supply chain risk assessment with dual generalized triangular fuzzy Bonferroni mean operators", Journal of Intelligent & Fuzzy Systems, 37(2), pp. 1953-1965 (2019).
37. Li, Z. and Wei, G. "Pythagorean fuzzy heronian mean operators in multiple attribute decision making and their  application to supplier selection", International Journal of Knowledge-Based and Intelligent Engineering Systems, 23(2), pp. 77-91 (2019).
38. Ayub, S., Abdullah, S., Ghani, F., et al. "Cubic fuzzy Heronian mean Dombi aggregation operators and their application on multi-attribute decision-making problem", Soft Computing, 25(6), pp. 4175-4189 (2021).
39. Liu, P.D., Liu, J.L., and Merigo, J.M. "Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making", Applied Soft Computing, 62(10), pp. 395-422 (2018).
40. Peng, J.J., Wang, J.Q., Hu, J.H., et al. "Multicriteria decision-making approach based on singlevalued neutrosophic hesitant fuzzy geometric weighted choquet integral heronian mean operator", Journal of Intelligent & Fuzzy Systems, 35(3), pp. 1-14 (2018).
41. Liu, P. and Zhang, L. "Multiple criteria decision making method based on neutrosophic hesitant fuzzy Heronian mean aggregation operators", Journal of Intelligent & Fuzzy Systems, 32(1), pp. 303-319 (2017).
42. Liu, P.D. "Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators", Computers & Industrial Engineering, 108, pp. 199-212 (2017).
43. Sellak, H., Ouhbi, B., Frikh, B., et al. "Expertise-based consensus building for MCGDM with hesitant fuzzy linguistic information", Information Fusion, 50, pp. 54-70 (2019).
44. Brauers, W.K.M. and Zavadskas, E.K. "Project management by multimoora as an instrument for transition economies", Ukio Technologinis Ir Ekonominis Vystymas, 16(1), pp. 5-24 (2010).
45. Hafezalkotob, A., Hafezalkotob, A., Liao, H.C., et al. "An overview of MULTIMOORA for multi-criteria decision-making: Theory, developments, applications, and challenges", Information Fusion, 51, pp. 145-177 (2019).
46. Zhao, H., You, J.X., and Liu, H.C. "Failure mode and effect analysis using MULTIMOORA method with continuous weighted entropy under interval-valued intuitionistic fuzzy environment", Soft Computing, 21(18), pp. 5355-5367 (2016).
47. Liu, H.C., Zhao, H., You, X.Y., et al. "Robot Evaluation and selection using the hesitant fuzzy linguistic MULTIMOORA method", Journal of Testing and Evaluation, 47(2), pp. 1405-1426 (2019).
48. Tian, Z.P., Wang, J., Wang, J.Q., et al. "An improved MULTIMOORA approach for multi-criteria decisionmaking based on interdependent inputs of simplified neutrosophic linguistic information", Neural Computing & Applications, 28, pp. S585-S597 (2017).
49. Nabeeh, N.A., Abdel-Monem, A., and Abdelmouty, A. "A hybrid approach of neutrosophic with MULTIMOORA in application of personnel selection", Neutrosophic Sets and Systems, 30, pp. 1-21 (2019).
50. Mi, X.M., Liao, H.C., Liao, Y., et al. "Green suppler selection by an integrated method with stochastic acceptability analysis and MULTIMOORA", Technological and Economic Development of Economy, 26(3), pp. 549-572 (2020).
51. Wu, S.M., You, X.Y., Liu, H.C., et al. "Improving quality function deployment analysis with the cloud MULTIMOORA method", International Transactions in Operational Research, 27(3), pp. 1600-1621 (2020).
52. Ji, P., Zhang, H.Y., and Wang, J.Q. "A fuzzy decision support model with sentiment analysis for items comparison in e-commerce: The case study of PConline. com", IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(10), pp. 1993-2004 (2018).
53. Liu, P., Cheng, S., and Zhang, Y. "An extended multicriteria group decision-making PROMETHEE method based on probability multi-valued neutrosophic sets", International Journal of Fuzzy Systems, 21(2), pp. 388-406 (2019).
54. Liu, P.D., Cheng, S.F., and Zhang, Y.M. "An extended multi-criteria group decision-making PROMETHEE method based on probability multi-valued neutrosophic sets", International  Journal of Fuzzy Systems, 21(2), pp. 388-406 (2019).
55. Liu, P. and Li, Y. "An extended MULTIMOORA method for probabilistic linguistic multi-criteria group decision-making based on prospect theory", Computers & Industrial Engineering, 136, pp. 528-545 (2019).
56. Liu, P. and Chen, S.M. "Group decision making based on heronian aggregation operators of intuitionistic fuzzy numbers", IEEE Trans Cybern, 47(9), pp. 2514- 2530 (2017).
57. Sumi, H. "The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics", Advances in Mathematics, 290, pp. 809-859 (2014). 
58. Wang, T.X., Li, H.X., Zhang, L.B., et al. "A three-way decision model based on cumulative prospect theory", Information Sciences, 519, pp. 74-92 (2020).
59. Ghanbaripour, A.N., Sher, W., and Yousefi, A. "Critical success factors for subway construction projects - main contractors' perspectives", International Journal of Construction Management, 20(3), pp. 177-195 (2020).
60. Nouri, F., Khorasani-Zavareh, D., and Mohammadi, R. "Factor's affecting safe emergency evacuation of subways in Iran: findings of a qualitative study",Journal of Injury &  Violence Research, 12(2), pp. 117- 134 (2020).
61. Kahneman, D. and Tversky, K.A. "Prospect theory: An analysis of decision under risk", Econometrica, 47(2), pp. 263-291 (1979).
62. Li, Y.H., Liu, P.D., and Chen, Y.B. "Some single valued neutrosophic number heronian mean operators and their application in multiple attribute group decision making", Informatica, 27(1), pp. 85-110 (2016).
63. Biswas, P., Pramanik, S., and Giri, B.C. "TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment", Neural Computing & Applications, 27(3), pp. 727-737 (2016).
64. Ji, P., Zhang, H.Y., and Wang, J.Q. "A projectionbased TODIM method under multi-valued neutrosophic environments and its application in personnel selection", Neural Computing & Applications, 29(1), pp. 1-14 (2017).