Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection
Abstract
:1. Introduction
2. Preliminaries
2.1. Pythagorean Fuzzy Sets
- (1)
- If then
- (2)
- If
- (1)
- (2)
- (3)
- (4)
- (5)
- (1)
- (2)
- (3)
- (4)
- (5)
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- (5)
- (6)
- (7)
- .
2.2. HM Operator
- (i)
- when
- (ii)
- when
- (iii)
- when
- (1)
- When , it becomes the arithmetic mean operator.
- (2)
- When it becomes the geometric mean operator.
3. The HM Operators for PFNs
3.1. The PFHM Operator
- (1)
- First of all, we prove (7) is kept.Therefore,Moreover,Furthermore,
- (2)
- Next, we prove (7) is a PFN. Let
- Since we can getSimilarly, we can get
- Obviously, then
- If then we can get .
- If then
3.2. The WPFHM Operator
- (1)
- First of all, we prove that (19) and (20) are kept. For the first case, when , according to the operational laws of PFNs, we getThereafter,Moreover,Therefore,For the second case, when , we getThen,
- (2)
- Next, we prove the (19) and (20) are PFNs. For the first case, when
- Since , we can getTherefore,Similarly, we can getTherefore,
- Since , we can get the following inequality:
- (1)
- For the first case, when .Since we can get
- (2)
- For the second case, when
- (1)
- If then we can get .
- (2)
- If then
3.3. The PFDHM Operator
- (1)
- First of all, we prove (35) is kept.Then,Thereafter,Furthermore,
- (2)
- Next, we prove (34) is a PFN.Let
- Since we can getTherefore similarly, we can get
- Obviously, then
- If then we can get
- If then
3.4. The WPFDHM Operator
- (1)
- First of all, we prove that (45) and (46) are kept.For the first case, when , we getThen,Thereafter,Furthermore,For the second case, when , we getThen,
- (2)
- Next, we prove the (45) and (46) are PFNs. For the first case, when
- Since , we can getTherefore, Similarly, we can getTherefore, .
- Since , we can get the following inequality:
- (1)
- For the first case, when .Since we can get
- (2)
- For the second case, when
- (1)
- If then we can get .
- (2)
- If then
4. A MAGDM Approach Based on the Proposed PFHM Operator
5. An Illustrate Example
5.1. Decision-Making Processes
5.2. Comparative Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Yager, R.R. Pythagorean fuzzy subsets. In Proceedings of the Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, AB, Canada, 24–28 June 2013; pp. 57–61. [Google Scholar]
- Yager, R.R. Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 2014, 22, 958–965. [Google Scholar] [CrossRef]
- Zhang, X.L.; Xu, Z.S. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int. J. Intell. Syst. 2014, 29, 1061–1078. [Google Scholar] [CrossRef]
- Peng, X.; Yang, Y. Some results for Pythagorean Fuzzy Sets. Int. J. Intell. Syst. 2015, 30, 1133–1160. [Google Scholar] [CrossRef]
- Reformat, M.Z.; Yager, R.R. Suggesting Recommendations Using Pythagorean Fuzzy Sets illustrated Using Netflix Movie Data. In Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Montpellier, France, 15–19 July 2014; pp. 546–556. [Google Scholar]
- Gou, X.; Xu, Z.; Ren, P. The Properties of Continuous Pythagorean Fuzzy Information. Int. J. Intell. Syst. 2016, 31, 401–424. [Google Scholar] [CrossRef]
- Garg, H. A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making. Int. J. Intell. Syst. 2016, 31, 886–920. [Google Scholar] [CrossRef]
- Zeng, S.; Chen, J.; Li, X. A Hybrid Method for Pythagorean Fuzzy Multiple-Criteria Decision Making. Int. J. Inf. Technol. Decis. Making 2016, 15, 403–422. [Google Scholar] [CrossRef]
- Wei, G.W. Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 2119–2132. [Google Scholar] [CrossRef]
- Gao, H.; Lu, M.; Wei, G.W.; Wei, Y. Some novel Pythagorean fuzzy interaction aggregation operators in multiple attribute decision making. Fundam. Inf. 2018, 159, 385–428. [Google Scholar] [CrossRef]
- Ren, P.J.; Xu, Z.S.; Gou, X.J. Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl. Soft Comput. 2016, 42, 246–259. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M. Pythagorean Fuzzy Maclaurin Symmetric Mean Operators in multiple attribute decision making. Int. J. Intell. Syst. 2018, 33, 1043–1070. [Google Scholar] [CrossRef]
- Maclaurin, C. A second letter to Martin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra. Philos. Trans. R. Soc. Lond. Ser. A 1729, 36, 59–96. [Google Scholar]
- Wu, S.J.; Wei, G.W. Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Int. J. Knowl. Based Intell. Eng. Syst. 2017, 21, 189–201. [Google Scholar] [CrossRef]
- Wei, G.W.; Wei, Y. Similarity measures of Pythagorean fuzzy sets based on cosine function and their applications. Int. J. Intell. Syst. 2018, 33, 634–652. [Google Scholar] [CrossRef]
- Wei, G.W.; Gao, H. The generalized Dice similarity measures for picture fuzzy sets and their applications. Informatica 2018, 29, 107–124. [Google Scholar] [CrossRef]
- Wei, G.W. Some similarity measures for picture fuzzy sets and their applications. Iran. J. Fuzzy Syst. 2018, 15, 77–89. [Google Scholar]
- Wei, G.W. Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Informatica 2017, 28, 547–564. [Google Scholar] [CrossRef]
- Xue, W.T.; Xu, Z.S.; Zhang, X.L.; Tian, X.L. Pythagorean Fuzzy LINMAP Method Based on the Entropy Theory for Railway Project Investment Decision Making. Int. J. Intell. Syst. 2018, 33, 93–125. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M. Pythagorean fuzzy power aggregation operators in multiple attribute decision making. Int. J. Intell. Syst. 2018, 33, 169–186. [Google Scholar] [CrossRef]
- Wan, S.-P.; Jin, Z.; Dong, J.-Y. Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees. Knowl. Inf. Syst. 2018, 55, 437–466. [Google Scholar] [CrossRef]
- Baloglu, U.B.; Demir, Y. An Agent-Based Pythagorean Fuzzy Approach for Demand Analysis with Incomplete Information. Int. J. Intell. Syst. 2018, 33, 983–997. [Google Scholar] [CrossRef]
- Liang, D.C.; Zhang, Y.R.J.; Xu, Z.S.; Darko, A.P. Pythagorean fuzzy Bonferroni mean aggregation operator and its accelerative calculating algorithm with the multithreading. Int. J. Intell. Syst. 2018, 33, 615–633. [Google Scholar] [CrossRef]
- Wei, G.W.; Zhang, Z.P. Some Single-Valued Neutrosophic Bonferroni Power Aggregation Operators in Multiple Attribute Decision Making. J. Ambient Intell. Humaniz. Comput. 2018. [Google Scholar] [CrossRef]
- Wang, J.; Wei, G.W.; Wei, Y. Models for Green Supplier Selection with Some 2-Tuple Linguistic Neutrosophic Number Bonferroni Mean Operators. Symmetry 2018, 10, 131. [Google Scholar] [CrossRef]
- Wei, G.W. Picture uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Kybernetes 2017, 46, 1777–1800. [Google Scholar] [CrossRef]
- Wei, G.W. Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making. Int. J. Fuzzy Syst. 2017, 19, 997–1010. [Google Scholar] [CrossRef]
- Jiang, X.P.; Wei, G.W. Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2014, 27, 2153–2162. [Google Scholar]
- Mandal, P.; Ranadive, A.S. Decision-theoretic rough sets under Pythagorean fuzzy information. Int. J. Intell. Syst. 2018, 33, 818–835. [Google Scholar] [CrossRef]
- Chen, T.-Y. An Interval-Valued Pythagorean Fuzzy Outranking Method with a Closeness-Based Assignment Model for Multiple Criteria Decision Making. Int. J. Intell. Syst. 2018, 33, 126–168. [Google Scholar] [CrossRef]
- Garg, H. A Linear Programming Method Based on an Improved Score Function for Interval-Valued Pythagorean Fuzzy Numbers and Its Application to Decision-Making. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 2018, 26, 67–80. [Google Scholar] [CrossRef]
- Khan, M.S.A.; Abdullah, S.; Ali, M.Y.; Hussain, I.; Farooq, M. Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment. J. Intell. Fuzzy Syst. 2018, 34, 267–282. [Google Scholar] [CrossRef]
- Garg, H. New exponential operational laws and their aggregation operators for interval-valued Pythagorean fuzzy multicriteria decision-making. Int. J. Intell. Syst. 2018, 33, 653–683. [Google Scholar] [CrossRef]
- Li, D.Q.; Zeng, W.Y. Distance Measure of Pythagorean Fuzzy Sets. Int. J. Intell. Syst. 2018, 33, 348–361. [Google Scholar] [CrossRef]
- Gao, H. Pythagorean Fuzzy Hamacher Prioritized Aggregation Operators in Multiple Attribute Decision Making. J. Intell. Fuzzy Syst. 2018, 35, 2229–2245. [Google Scholar] [CrossRef]
- Wei, G.; Wei, Y. Some single-valued neutrosophic dombi prioritized weighted aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2018, 35, 2001–2013. [Google Scholar] [CrossRef]
- Gao, H.; Wei, G.W.; Huang, Y.H. Dual hesitant bipolar fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making. IEEE Access 2018, 6, 11508–11522. [Google Scholar] [CrossRef]
- Ran, L.G.; Wei, G.W. Uncertain prioritized operators and their application to multiple attribute group decision making. Technol. Econ. Dev. Econ. 2015, 21, 118–139. [Google Scholar] [CrossRef]
- Zhao, X.F.; Li, Q.X.; Wei, G.W. Some prioritized aggregating operators with linguistic information and their application to multiple attribute group decision making. J. Intell. Fuzzy Syst. 2014, 26, 1619–1630. [Google Scholar]
- Zhou, L.Y.; Lin, R.; Zhao, X.F.; Wei, G.W. Uncertain linguistic prioritized aggregation operators and their application to multiple attribute group decision making. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 2013, 21, 603–627. [Google Scholar] [CrossRef]
- Lin, R.; Zhao, X.F.; Wei, G.W. Fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2013, 24, 879–888. [Google Scholar]
- Wei, G.W.; Lu, M. Dual hesitant Pythagorean fuzzy Hamacher aggregation operators in multiple attribute decision making. Arch. Control Sci. 2017, 27, 365–395. [Google Scholar] [CrossRef]
- Lu, M.; Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Hesitant pythagorean fuzzy hamacher aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 1105–1117. [Google Scholar] [CrossRef]
- Wei, G.W.; Lu, M.; Tang, X.Y.; Wei, Y. Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making. J. Intell. Fuzzy Syst. 2018, 33, 1197–1233. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Bipolar fuzzy Hamacher aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2018, 20, 1–12. [Google Scholar] [CrossRef]
- Wei, G.W. Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fundam. Inf. 2018, 157, 271–320. [Google Scholar] [CrossRef]
- Zhou, L.Y.; Zhao, X.F.; Wei, G.W. Hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2014, 26, 2689–2699. [Google Scholar]
- Wei, G.W.; Lu, M.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 1129–1142. [Google Scholar] [CrossRef]
- Tang, X.Y.; Wei, G.W. Models for green supplier selection in green supply chain management with Pythagorean 2-tuple linguistic information. IEEE Access 2018, 6, 8042–8060. [Google Scholar] [CrossRef]
- Huang, Y.H.; Wei, G.W. TODIM Method for Pythagorean 2-tuple Linguistic Multiple Attribute Decision Making. J. Intell. Fuzzy Syst. 2018, 35, 901–915. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure. Int. J. Fuzzy Syst. 2017, 19, 607–614. [Google Scholar] [CrossRef]
- Wei, G.W.; Gao, H.; Wang, J.; Huang, Y.H. Research on Risk Evaluation of Enterprise Human Capital Investment with Interval-valued bipolar 2-tuple linguistic Information. IEEE Access 2018, 6, 35697–35712. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft Comput. 2018, 22, 989–1002. [Google Scholar] [CrossRef]
- Wei, G.W. Interval-valued dual hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 1881–1893. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Hesitant bipolar fuzzy aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 1119–1128. [Google Scholar] [CrossRef]
- Wei, G.W. Picture fuzzy aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 713–724. [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, 1426–1458. [Google Scholar] [CrossRef]
- Hara, T.; Uchiyama, M.; Takahasi, S.E. A refinement of various mean inequalities. J. Inequal. Appl. 1998, 2, 387–395. [Google Scholar] [CrossRef]
- Wu, S.; Wang, J.; Wei, G.; Wei, Y. Research on Construction Engineering Project Risk Assessment with Some 2-Tuple Linguistic Neutrosophic Hamy Mean Operators. Sustainability 2018, 10, 1536. [Google Scholar] [CrossRef]
- Ma, Z.M.; Xu, Z.S. Symmetric Pythagorean Fuzzy Weighted Geometric_Averaging Operators and Their Application in Multicriteria Decision-Making Problems. Int. J. Intell. Syst. 2016, 31, 1198–1219. [Google Scholar] [CrossRef]
- Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Projection models for multiple attribute decision making with picture fuzzy information. Int. J. Mach. Learn. Cybern. 2018, 9, 713–719. [Google Scholar] [CrossRef]
- Lu, M.; Wei, G.W.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Bipolar 2-tuple linguistic aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 2017, 33, 1197–1207. [Google Scholar] [CrossRef]
- Merigo, J.M.; Gil-Lafuente, A.M. Fuzzy induced generalized aggregation operators and its application in multi-person decision making. Expert Syst. Appl. 2011, 38, 9761–9772. [Google Scholar] [CrossRef]
- Chen, T.Y. Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf. Fusion 2018, 41, 129–150. [Google Scholar] [CrossRef]
Ranking | ||
---|---|---|
Ranking | ||
---|---|---|
Ordering | |
---|---|
PFWA | |
PFWG |
© 2018 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
Li, Z.; Wei, G.; Lu, M. Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection. Symmetry 2018, 10, 505. https://doi.org/10.3390/sym10100505
Li Z, Wei G, Lu M. Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection. Symmetry. 2018; 10(10):505. https://doi.org/10.3390/sym10100505
Chicago/Turabian StyleLi, Zengxian, Guiwu Wei, and Mao Lu. 2018. "Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection" Symmetry 10, no. 10: 505. https://doi.org/10.3390/sym10100505