A Multi-Attribute Decision-Making Algorithm Using Q-Rung Orthopair Power Bonferroni Mean Operator and Its Application
Abstract
:1. Introduction
2. Preliminaries Knowledge
- (1)
- =()
- (2)
- = ()
- (3)
- (4)
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- (1)
- If, then
- (2)
- If, and if, then; if, then
3. q-ROFPBM Operator
3.1. The Concept and Demonstration of q-ROFPBM Operator
- (1)
- (2)
- (3)
- if and only if.
- (4)
- If, then, whereis the distance betweenand.
- (1)
- (2)
- (3)
- if and only if,
- (4)
- If, then, whereis the distance betweenand.
- (1)
- If n = 2, we can getTherefore, we know that if n = 2, the Definition 7, that is
- (2)
- If n = k (k > 2), that isIf n = k + 1, then, we can getFurther, we can infer thatThen, we can getAgain, according to the Formula (3) in Definition 2, we can getAgain, based on the Formula (4) in Definition 2, we can obtain
3.2. Some Properties of the q-ROFPBM Operator
- (1)
- (Idempotence). Suppose is a q-ROF number, and , then
- (2)
- (Permutation invariability). Suppose is a q-ROF number, and is any permutation and combination of , then
- (3)
- (Boundedness). Suppose is a q-ROF number. Let , , , then
4. The MADM Algorithm Based on the q-ROFPBM Operator
5. Numerical Example
5.1. Decision Process
5.2. Comparative Analysis
- (1)
- When using the q-ROFPA operator.
- (2)
- When using the q-ROFBM operator.
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Yu, C.; Shao, Y.; Wang, K.; Zhang, L. A group decision making sustainable supplier selection approach using extended TOPSIS under interval-valued Pythagorean fuzzy environment. Expert Syst. Appl. 2019, 121, 1–17. [Google Scholar] [CrossRef]
- Awasthi, A.; Govindan, K.; Gold, S. Multi-tier sustainable global supplier selection using a fuzzy AHP-VIKOR based approach. Int. J. Prod. Econ. 2018, 195, 106–117. [Google Scholar] [CrossRef]
- Wang, L.; Zhang, H.Y.; Wang, J.Q.; Li, L. Picture fuzzy normalized projection based VIKOR method for the risk evaluation of construction project. Appl. Soft Comput. 2018, 64, 216–226. [Google Scholar] [CrossRef]
- Deng, X.M.; Wei, G.W.; Gao, H.; Wang, J. Models for safety assessment of construction project with some 2-tuple linguistic Pythagorean fuzzy Bonferroni mean operators. IEEE Access 2018, 6, 52105–52137. [Google Scholar] [CrossRef]
- Yang, Z.L.; Ouyang, T.; Fu, X.; Peng, X. A decision-making algorithm for online shopping using deep-learning–based opinion pairs mining and q-rung orthopair fuzzy interaction Heronian mean operators. Int. J. Intell. Syst. 2020, 35, 783–825. [Google Scholar] [CrossRef]
- Yang, Z.L.; Xiong, G.M.; Cao, Z.H.; Li, Y.C.; Huang, L.C. A decision method for online purchases considering dynamic information preference based on sentiment orientation classification and discrete DIFWA operators. IEEE Access 2019, 7, 77008–77026. [Google Scholar] [CrossRef]
- Hao, Z.N.; Xu, Z.S.; Zhao, H.; Fujita, H. A dynamic weight determination approach based on the intuitionistic fuzzy bayesian network and its application to emergency decision making. IEEE Trans. Fuzzy Syst. 2018, 26, 1893–1907. [Google Scholar] [CrossRef]
- Ding, X.F.; Liu, H.C.; Shi, H. A dynamic approach for emergency decision making based on prospect theory with interval-valued Pythagorean fuzzy linguistic variables. Comput. Ind. Eng. 2019, 131, 57–65. [Google Scholar] [CrossRef]
- Yang, Z.L.; Li, J.Q.; Huang, L.C.; Shi, Y.Y. Developing dynamic intuitionistic normal fuzzy aggregation operators for multi-attribute decision-making with time sequence preference. Expert Syst. Appl. 2017, 82, 344–356. [Google Scholar] [CrossRef]
- Wan, S.P.; Xu, G.L.; Wang, F.; Dong, J.Y. A new method for Atanassov’s interval-valued intuitionistic fuzzy MAGDM with incomplete attribute weight information. Inform. Sci. 2015, 316, 329–347. [Google Scholar] [CrossRef]
- Liu, P.D.; Chen, S.M. Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Inform. Sci. 2018, 430, 599–619. [Google Scholar] [CrossRef]
- Wan, S.P.; Li, S.Q.; Dong, J.Y. A three-phase method for Pythagorean fuzzy multi-attribute group decision making and application to haze management. Comput. Ind. Eng. 2018, 123, 348–363. [Google Scholar] [CrossRef]
- Hussian, Z.; Yang, M.S. Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS. Int. J. Intell. Syst. 2019, 34, 2633–2654. [Google Scholar] [CrossRef]
- Yang, Z.L.; Chang, J.P. Interval-valued Pythagorean normal fuzzy information aggregation operators for multi-attribute decision making. IEEE Access 2020, 8, 51295–51314. [Google Scholar] [CrossRef]
- Jana, C.; Senapati, T.; Pal, M.; Yager, R.R. Picture fuzzy Dombi aggregation operators: Application to MADM process. Appl. Soft Comput. 2019, 74, 99–109. [Google Scholar] [CrossRef]
- Ju, Y.B.; Ju, D.W.; Gonzalez, E.; Giannakis, M.; Wang, A.H. Study of site selection of electric vehicle charging station based on extended GRP method under picture fuzzy environment. Comput. Ind. Eng. 2019, 135, 1271–1285. [Google Scholar] [CrossRef]
- Mahmood, T.; Ullah, K.; Khan, Q.; Jan, N. An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput. Appl. 2018, 31, 7041–7053. [Google Scholar] [CrossRef]
- Sarkar, A.; Biswas, A. Multicriteria decision-making using Archimedean aggregation operators in pythagorean hesitant fuzzy environment. Int. J. Intell. Syst. 2019, 34, 1361–1386. [Google Scholar] [CrossRef]
- Yang, Z.L.; Lin, X.; Garg, H.; Meng, Q. Decision support algorithm for selecting an antivirus mask over COVID-19 pandemic under spherical normal fuzzy environment. Int. J. Environ. Res. Public Health 2020, 17, 3407. [Google Scholar] [CrossRef]
- Yang, Z.L.; Lin, X.; Cao, Z.H.; Yang, Z.L.; Lin, X.; Li, J.Q. Q-rung orthopair normal fuzzy aggregation operators and their application in multi-attribute decision-making. Mathematics 2019, 7, 1142. [Google Scholar] [CrossRef] [Green Version]
- Yang, Z.L.; Garg, H.; Li., J.; Garg, H.; Srivastavad, G.; Garg, H.; Cao, Z. Investigation of multiple heterogeneous relationships using a q-rung orthopair fuzzy multi-criteria decision algorithm. Neural Comput. Appl. 2020, 1–22. [Google Scholar] [CrossRef]
- Ye, J.M.; Ai, Z.H.; Xu, Z.S. Single variable differential calculus under q-rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application. Int. J. Intell. Syst. 2019, 34, 1387–1415. [Google Scholar] [CrossRef]
- Darko, A.P.; Liang, D. Some q-rung orthopair fuzzy Hamacher aggregation operators and their application to multiple attribute group decision making with modified EDAS method. Eng. Appl. Artif. Intll. 2020, 87, 103259. [Google Scholar] [CrossRef]
- Yager, R.R. Generalized Orthopair Fuzzy Sets. IEEE Trans. Fuzzy Syst. 2017, 25, 1222–1230. [Google Scholar] [CrossRef]
- Liu, P.D.; Liu, J.L. Some q-rung orthopair fuzzy aggregation operators and their applications to multi-attribute group decision making. Int. J. Intell. Syst. 2018, 33, 316–324. [Google Scholar]
- Ju, Y.; Luo, C.; Ma, J.; Gao, H.X.; Gao, H.X.; Santibanez Gonzalez, E.D.R.; Wang, A. Some interval-valued q-rung orthopair weighted averaging operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst. 2019, 34, 2584–2606. [Google Scholar] [CrossRef]
- Liu, P.D.; Liu, J.L. Some q-rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making. Int. J. Intell. Syst. 2018, 33, 315–347. [Google Scholar] [CrossRef]
- Du, W.S. Minkowski-type distance measures for generalized orthopair fuzzy sets. Int. J. Intell. Syst. 2018, 33, 801–806. [Google Scholar] [CrossRef]
- Wei, G.W.; Gao, H.; Wei, Y. Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. Int. J. Intell. Syst. 2018, 33, 1426–1432. [Google Scholar] [CrossRef]
- Peng, X.D.; Dai, J.G.; Gag, H. Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function. Int. J. Intell. Syst. 2018, 33, 2252–2282. [Google Scholar] [CrossRef]
- Liu, P.D.; Wang, Y.M. Multiple attribute decision making based on q-rung orthopair fuzzy generalized Maclaurin symmetic mean operators. Inform. Sci. 2020, 518, 181–210. [Google Scholar] [CrossRef]
- Garg, H.; Chen, S.M. Multiattribute group decision making based on neutrality aggregation operators of q-rung orthopair fuzzy sets. Inform. Sci. 2020, 517, 427–447. [Google Scholar] [CrossRef]
- Garg, H. A novel trigonometric operation-based q-rung orthopair fuzzy aggregation operator and its fundamental properties. Neural Comput. Appl. 2020, 31, 2223–2586. [Google Scholar] [CrossRef]
- Shao, Y.B.; Zhuo, J.L. Basic theory of line integrals under the q-rung orthopair fuzzy environment and their applications. Int. J. Intell. Syst. 2020, 15, 1453–1545. [Google Scholar] [CrossRef]
- Pinar, A.; Boran, F.E. A q-rung orthopair fuzzy multi-criteria group decision making method for supplier selection based on a novel distance measure. Int. J. Mach. Learn. Cyb. 2020. online. [Google Scholar] [CrossRef]
- Liu, P.; Wang, P. Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst. 2018, 33, 259–280. [Google Scholar] [CrossRef]
- Liu, P.; Liu, X. Multiattribute group decision making methods based on linguistic Intuitionistic fuzzy power Bonferroni mean operators. Complexity 2017, 2017, 3571459. [Google Scholar] [CrossRef]
0.4 | 0.5 | 0.5 | 0.4 | 0.2 | 0.7 | 0.2 | 0.5 | |
0.6 | 0.4 | 0.6 | 0.3 | 0.6 | 0.3 | 0.3 | 0.6 | |
0.5 | 0.5 | 0.4 | 0.5 | 0.4 | 0.4 | 0.5 | 0.4 | |
0.7 | 0.2 | 0.5 | 0.4 | 0.2 | 0.5 | 0.6 | 0.7 | |
0.5 | 0.3 | 0.3 | 0.4 | 0.6 | 0.2 | 0.4 | 0.4 |
1 | 0.352 | 0.536 | 0.549 | 0.414 | 0.453 | 0.453 | 0.540 | 0.483 | 0.464 | 0.335 |
2 | 0.389 | 0.532 | 0.571 | 0.411 | 0.458 | 0.453 | 0.574 | 0.477 | 0.485 | 0.335 |
3 | 0.408 | 0.528 | 0.581 | 0.408 | 0.464 | 0.452 | 0.590 | 0.470 | 0.499 | 0.334 |
1 | 0.015 | 0.788 | 0.089 | 0.633 | 0.042 | 0.686 | 0.086 | 0.726 | 0.046 | 0.524 |
2 | 0.000 | 0.931 | 0.003 | 0.800 | 0.000 | 0.854 | 0.004 | 0.883 | 0.001 | 0.685 |
3 | 0.000 | 0.986 | 0.000 | 0.911 | 0.000 | 0.950 | 0.000 | 0.964 | 0.000 | 0.814 |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
He, P.; Yang, Z.; Hou, B. A Multi-Attribute Decision-Making Algorithm Using Q-Rung Orthopair Power Bonferroni Mean Operator and Its Application. Mathematics 2020, 8, 1240. https://doi.org/10.3390/math8081240
He P, Yang Z, Hou B. A Multi-Attribute Decision-Making Algorithm Using Q-Rung Orthopair Power Bonferroni Mean Operator and Its Application. Mathematics. 2020; 8(8):1240. https://doi.org/10.3390/math8081240
Chicago/Turabian StyleHe, Ping, Zaoli Yang, and Bowen Hou. 2020. "A Multi-Attribute Decision-Making Algorithm Using Q-Rung Orthopair Power Bonferroni Mean Operator and Its Application" Mathematics 8, no. 8: 1240. https://doi.org/10.3390/math8081240