Convex Aggregation Operators and Their Applications to Multi-Hesitant Fuzzy Multi-Criteria Decision-Making
Abstract
:1. Introduction
2. Hesitant Fuzzy Sets and Multi-Hesitant Fuzzy Sets
- (1)
- ;
- (2)
- ();
- (3)
- ;
- (4)
- .
- (1)
- if, then;
- (2)
- if, then:
- -
- if, then;
- -
- if, then;
- -
- if, then.
3. The Convex Combination Operation and Some Aggregation Operators of MHFNs
- (1)
- (Monotonicity) Letbe a collection of MHFNs. If for all,, then
- (2)
- (Commutativity) Ifis a permutation of, then
- (3)
- (Boundedness) Ifand, whereand, then
- (1)
- (Monotonicity) Letbe a collection of MHFNs. If for all,, then
- (2)
- (Commutativity) Ifis a permutation of, then
- (3)
- (Boundedness) Ifand, whereand, then
- (1)
- (Monotonicity) Letbe a collection of MHFNs. If for all,, then
- (2)
- (Commutativity) Ifis a permutation of, then
- (3)
- (Boundedness) Ifand, whereand, then
- (1)
- (Monotonicity) Letbe a collection of MHFNs. If for all,, then
- (2)
- (Commutativity) Ifis a permutation of, then
- (3)
- (Boundedness) Ifand, whereand, then
4. The MCDM Method Based on Aggregation Operators with MHFNs
5. An Illustrative Example
5.1. An Illustration of the Proposed Approach
5.2. Sensitivity Analysis
5.3. A Comparison Analysis and Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst. 2010, 25, 529–539. [Google Scholar] [CrossRef]
- Torra, V.; Narukawa, Y. On hesitant fuzzy sets and decision. In Proceedings of the 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 20–24 August 2009; pp. 1378–1382. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–356. [Google Scholar] [CrossRef]
- Song, C.; Xu, Z.; Zhao, H. A novel comparison of probabilistic hesitant fuzzy elements in multi-criteria decision making. Symmetry 2018, 10, 177. [Google Scholar] [CrossRef]
- Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry 2017, 9, 136. [Google Scholar] [CrossRef]
- Faizi, S.; Rashid, T.; Saabun, W.; Zafar, S.; Wtróbski, J. Decision making with uncertainty using hesitant fuzzy sets. Int. J. Fuzzy Syst. 2018, 20, 93–103. [Google Scholar] [CrossRef]
- Liao, H.; Wu, D.; Huang, Y.; Ren, P.; Xu, Z.; Verma, M. Green logistic provider selection with a hesitant fuzzy linguistic thermodynamic method integrating cumulative prospect theory and PROMETHEE. Sustainability 2018, 10, 1291. [Google Scholar] [CrossRef]
- Liu, P.; Gao, H. Multi-criteria decision making based on generalized Maclaurin symmetric means with multi-hesitant fuzzy linguistic information. Symmetry 2018, 10, 81. [Google Scholar] [CrossRef]
- Xia, M.M.; Xu, Z.S. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 2011, 52, 395–407. [Google Scholar] [CrossRef] [Green Version]
- Zhu, B.; Xu, Z.S.; Xia, M.M. Hesitant fuzzy geometric Bonferoni means. Inf. Sci. 2012, 205, 72–85. [Google Scholar] [CrossRef]
- Wei, G.W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl. Based Syst. 2012, 31, 176–182. [Google Scholar] [CrossRef]
- Xia, M.M.; Xu, Z.S.; Chen, N. Some Hesitant fuzzy aggregation operators with their application in group decision making. Group Decis. Negot. 2013, 22, 259–279. [Google Scholar] [CrossRef]
- Zhang, Z.M.; Wang, C.; Tian, D.Z.; Li, K. Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making. Comput. Ind. Eng. 2014, 67, 116–138. [Google Scholar] [CrossRef]
- Zhou, W. An Accurate method for determining hesitant fuzzy aggregation operator weights and its application to project investment. Int. J. Intell. Syst. 2014, 29, 668–686. [Google Scholar] [CrossRef]
- Zhang, Z.M. Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making. Inf. Sci. 2013, 234, 150–181. [Google Scholar] [CrossRef]
- Yu, D.J. Some hesitant fuzzy information aggregation operators based on Einstein operational laws. Int. J. Intell. Syst. 2014, 29, 320–340. [Google Scholar] [CrossRef]
- Chen, N.; Xu, Z.S.; Xia, M.M. Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 2013, 37, 2197–2211. [Google Scholar] [CrossRef]
- Xu, Z.S.; Xia, M.M. Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 2011, 181, 2128–2138. [Google Scholar] [CrossRef]
- Xu, Z.S.; Xia, M.M. On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 2011, 26, 410–425. [Google Scholar] [CrossRef]
- Farhadinia, B. Distance and similarity measures for higher order hesitant fuzzy sets. Knowl. Based Syst. 2014, 55, 43–48. [Google Scholar] [CrossRef]
- Wang, L.; Ni, M.F.; Yu, Z.K.; Zhu, L. Power geometric operators of hesitant multiplicative fuzzy numbers and their application to multiple attribute group decision making. Math. Probl. Eng. 2014, 2014, 186502. [Google Scholar] [CrossRef]
- Torres, R.; Salas, R.; Astudillo, H. Time-based hesitant fuzzy information aggregation approach for decision-making problems. Int. J. Intell. Syst. 2014, 29, 579–595. [Google Scholar] [CrossRef]
- Qian, G.; Wang, H.; Feng, X.Q. Generalized hesitant fuzzy sets and their application in decision support system. Knowl. Based Syst. 2013, 37, 357–365. [Google Scholar] [CrossRef]
- Meng, F.Y.; Chen, X.H.; Zhang, Q. Induced generalized hesitant fuzzy Shapley hybrid operators and their application in multi-attribute decision making. Appl. Soft Comput. 2015, 28, 599–607. [Google Scholar] [CrossRef]
- Zhou, W.; Xu, Z.S. Optimal discrete fitting aggregation approach with hesitant fuzzy information. Knowl. Based Syst. 2015, 78, 22–33. [Google Scholar] [CrossRef]
- Tan, C.Q.; Yi, W.T.; Chen, X.H. Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making. Appl. Soft Comput. 2015, 26, 325–349. [Google Scholar] [CrossRef]
- Meng, F.Y.; Chen, X.H. Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures. Cognative Comput. 2015, 7, 445–463. [Google Scholar] [CrossRef]
- Liao, H.C.; Xu, Z.S.; Zeng, X.J. Novel correlation coefficients between hesitant fuzzy sets and their application in decision making. Knowl. Based Syst. 2015, 82, 115–127. [Google Scholar] [CrossRef]
- Li, D.Q.; Zeng, W.Y.; Li, J.H. New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Eng. Appl. Artif. Intell. 2015, 40, 11–16. [Google Scholar] [CrossRef]
- Hu, J.H.; Zhang, X.L.; Chen, X.H.; Liu, Y.M. Hesitant fuzzy information measures and their applications in multi-criteria decision making. Int. J. Syst. Sci. 2015, 87, 91–103. [Google Scholar] [CrossRef]
- Zhang, N.; Wei, G.W. Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Appl. Math. Model. 2013, 37, 4938–4947. [Google Scholar] [CrossRef]
- Zhang, X.L.; Xu, Z.S. The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment. Knowl. Based Syst. 2014, 61, 48–58. [Google Scholar] [CrossRef]
- Farhadinia, B. A novel method of ranking hesitant fuzzy values for multiple attribute decision-making problems. Int. J. Intell. Syst. 2013, 28, 752–767. [Google Scholar] [CrossRef]
- Farhadinia, B. Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 2013, 240, 129–144. [Google Scholar] [CrossRef]
- Peng, J.J.; Wang, J.Q.; Wang, J.; Yang, L.J.; Chen, X.H. An extension of ELECTRE to multi-criteria decision-making problems with multi-hesitant fuzzy sets. Inf. Sci. 2015, 307, 113–126. [Google Scholar] [CrossRef]
- Chen, N.; Xu, Z.S. Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems. Inf. Sci. 2015, 292, 175–197. [Google Scholar] [CrossRef]
- Yager, R.R. Prioritized aggregation operators. Int. J. Approx. Reason. 2008, 48, 263–274. [Google Scholar] [CrossRef]
- Wang, Z.; Klir, G.J. Fuzzy Measure Theory; Plenum Press: New York, NY, USA, 1992. [Google Scholar]
- Schmeidler, D. Subjective probability and expected utility without additivity. Econometrica 1989, 57, 517–587. [Google Scholar] [CrossRef]
{0.4, 0.5, 0.7} | {0.5, 0.5, 0.8} | {0.6, 0.6, 0.9} | {0.5, 0.6} | |
{0.6, 0.7, 0.8} | {0.5, 0.6} | {0.6, 0.7, 0.7} | {0.4, 0.5} | |
{0.6, 0.8} | {0.2, 0.3, 0.5} | {0.6, 0.6} | {0.5, 0.7} | |
{0.5, 0.5, 0.7} | {0.4, 0.5} | {0.8, 0.9} | {0.3, 0.4, 0.5} | |
{0.6, 0.7} | {0.5, 0.7} | {0.7, 0.8} | {0.3, 0.3, 0.4} |
Rankings | ||
---|---|---|
GMHFOWA | GMHFHWA | |
Rankings | ||
---|---|---|
GMHFPWA | GMHFCIWA | |
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Mei, Y.; Peng, J.; Yang, J. Convex Aggregation Operators and Their Applications to Multi-Hesitant Fuzzy Multi-Criteria Decision-Making. Information 2018, 9, 207. https://doi.org/10.3390/info9090207
Mei Y, Peng J, Yang J. Convex Aggregation Operators and Their Applications to Multi-Hesitant Fuzzy Multi-Criteria Decision-Making. Information. 2018; 9(9):207. https://doi.org/10.3390/info9090207
Chicago/Turabian StyleMei, Ye, Juanjuan Peng, and Junjie Yang. 2018. "Convex Aggregation Operators and Their Applications to Multi-Hesitant Fuzzy Multi-Criteria Decision-Making" Information 9, no. 9: 207. https://doi.org/10.3390/info9090207
APA StyleMei, Y., Peng, J., & Yang, J. (2018). Convex Aggregation Operators and Their Applications to Multi-Hesitant Fuzzy Multi-Criteria Decision-Making. Information, 9(9), 207. https://doi.org/10.3390/info9090207