The need for multi-attribute decision-making brings more and more complexity, and this type of decision-making extends to an ever wider range of areas of life. A recent model that captures many components of decision-making frameworks is the complex $ q $-rung picture fuzzy set (C$ q $-RPFS), a generalization of complex fuzzy sets and $ q $-rung picture fuzzy sets. From a different standpoint, linguistic terms are very useful to evaluate qualitative information without specialized knowledge. Inspired by the ease of use of the linguistic evaluations by means of 2-tuple linguistic term sets, and the broad scope of applications of C$ q $-RPFSs, in this paper we introduce the novel structure called 2-tuple linguistic complex $ q $-rung picture fuzzy sets (2TLC$ q $-RPFSs). We argue that this model prevails to represent the two-dimensional information over the boundary of C$ q $-RPFSs, thanks to the additional features of 2-tuple linguistic terms. Subsequently, some 2TLC$ q $-RPF aggregation operators are proposed. Fundamental cases include the 2TLC$ q $-RPF weighted averaging/geometric operators. Other sophisticated aggregation operators that we propose are based on the Hamacher operator. In addition, we investigate some essential properties of the new operators. These tools are the building blocks of a multi-attribute decision making strategy for problems posed in the 2TLC$ q $-RPFS setting. Furthermore, a numerical instance that selects an optimal machine is given to guarantee the applicability and effectiveness of the proposed approach. Finally, we conduct a comparison with other existing approaches.
Citation: Muhammad Akram, Ayesha Khan, Uzma Ahmad, José Carlos R. Alcantud, Mohammed M. Ali Al-Shamiri. A new group decision-making framework based on 2-tuple linguistic complex $ q $-rung picture fuzzy sets[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11281-11323. doi: 10.3934/mbe.2022526
The need for multi-attribute decision-making brings more and more complexity, and this type of decision-making extends to an ever wider range of areas of life. A recent model that captures many components of decision-making frameworks is the complex $ q $-rung picture fuzzy set (C$ q $-RPFS), a generalization of complex fuzzy sets and $ q $-rung picture fuzzy sets. From a different standpoint, linguistic terms are very useful to evaluate qualitative information without specialized knowledge. Inspired by the ease of use of the linguistic evaluations by means of 2-tuple linguistic term sets, and the broad scope of applications of C$ q $-RPFSs, in this paper we introduce the novel structure called 2-tuple linguistic complex $ q $-rung picture fuzzy sets (2TLC$ q $-RPFSs). We argue that this model prevails to represent the two-dimensional information over the boundary of C$ q $-RPFSs, thanks to the additional features of 2-tuple linguistic terms. Subsequently, some 2TLC$ q $-RPF aggregation operators are proposed. Fundamental cases include the 2TLC$ q $-RPF weighted averaging/geometric operators. Other sophisticated aggregation operators that we propose are based on the Hamacher operator. In addition, we investigate some essential properties of the new operators. These tools are the building blocks of a multi-attribute decision making strategy for problems posed in the 2TLC$ q $-RPFS setting. Furthermore, a numerical instance that selects an optimal machine is given to guarantee the applicability and effectiveness of the proposed approach. Finally, we conduct a comparison with other existing approaches.
| [1] |
L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
|
| [2] |
S. M. Chen, W. H. Hsiao, W. T. Jong, Bidirectional approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets Syst., 91 (1997), 339–353. https://doi.org/10.1016/S0165-0114(97)86594-3 doi: 10.1016/S0165-0114(97)86594-3
|
| [3] |
S. M. Chen, W. H. Hsiao, Bidirectional approximate reasoning for rule-based systems using interval-valued fuzzy sets, Fuzzy Sets Syst., 113 (2000), 185–203. https://doi.org/10.1016/S0165-0114(98)00351-0 doi: 10.1016/S0165-0114(98)00351-0
|
| [4] |
S. M. Chen, W. T. Jong, Fuzzy query translation for relational database systems, IEEE Trans. Syst. Man Cybern. Syst. Part B, 27 (1997), 714–721. https://doi.org/10.1109/3477.604117 doi: 10.1109/3477.604117
|
| [5] |
S. M. Chen, S. J. Niou, Fuzzy multiple-attributes group decision-making based on fuzzy preference relations, Expert Syst. Appl., 38 (2011), 3865–3872. https://doi.org/10.1016/j.eswa.2010.09.047 doi: 10.1016/j.eswa.2010.09.047
|
| [6] |
M. I. Ali, J. Zhan, M. J. Khan, T. Mahmood, H. Faizan, Another view on knowledge measures in atanassov intuitionistic fuzzy sets, Soft Comput., 26 (2022), 6507–6517. https://doi.org/10.1007/s00500-022-07127-3 doi: 10.1007/s00500-022-07127-3
|
| [7] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. https://doi.org/10.1007/978-3-7908-1870-3-1 doi: 10.1007/978-3-7908-1870-3-1
|
| [8] |
W. Wang, J. Zhan, J. Mi, A three-way decision approach with probabilistic dominance relations under intuitionistic fuzzy information, Inf. Sci., 582 (2022), 114–145. https://doi.org/10.1016/j.ins.2021.09.018 doi: 10.1016/j.ins.2021.09.018
|
| [9] | R. R. Yager, Pythagorean fuzzy subsets, in 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), (2013), 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 |
| [10] |
R. R. Yager, Pythagorean membership grades in multi-criteria decision making, IEEE Trans. Fuzzy Syst., 22 (2013), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
|
| [11] |
T. Senapati, R. R. Yager, Fermatean fuzzy sets, J. Ambient. Intell. Humaniz. Comput., 11 (2020), 663–674. https://doi.org/10.1007/s12652-019-01377-0 doi: 10.1007/s12652-019-01377-0
|
| [12] |
V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst., 25 (2010), 529–539. https://doi.org/10.1002/int.20418 doi: 10.1002/int.20418
|
| [13] |
W. Wang, X. Ma, Z. Xu, W. Pedrycz, J. Zhan, A three-way decision method with prospect theory to multi-attribute decision-making and its applications under hesitant fuzzy environments, Appl. Soft Comput., 126 (2022), 109283. https://doi.org/10.1016/j.asoc.2022.109283 doi: 10.1016/j.asoc.2022.109283
|
| [14] |
R. R. Yager, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst., 26 (2016), 1222–1230. https://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
|
| [15] |
B. C. Cuong, V. Kreinovich, Picture fuzzy sets, J. Comput. Sci. Cybern., 30 (2014), 409–420. https://doi.org/10.1109/WICT.2013.7113099 doi: 10.1109/WICT.2013.7113099
|
| [16] |
L. Li, R. T. Zhang, J. Wang, X. P. Shang, K. Y. Bai, A novel approach to multi-attribute group decision-making with $q$-rung picture linguistic information, Symmetry, 10 (2018), 172. https://doi.org/10.3390/sym10050172 doi: 10.3390/sym10050172
|
| [17] |
M. Akram, S. Alsulami, F. Karaaslan, A. Khan, $q$-Rung orthopair fuzzy graphs under Hamacher operators, J. Intell. Fuzzy Syst., 40 (2021), 1367–1390. https://doi.org/10.3233/JIFS-201700 doi: 10.3233/JIFS-201700
|
| [18] |
M. J. Khan, J. C. R. Alcantud, P. Kumam, W. Kumam, A. N. Al-Kenani, An axiomatically supported divergence measures for $q$-rung orthopair fuzzy sets, Int. J. Intell. Syst., 36 (2021), 6133–6155. https://doi.org/10.1002/int.22545 doi: 10.1002/int.22545
|
| [19] |
L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning Part Ⅰ, Inf. Sci., 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90046-8 doi: 10.1016/0020-0255(75)90046-8
|
| [20] |
H. Zhang, Linguistic intuitionistic fuzzy sets and application in MAGDM, J. Appl. Math., 2014 (2014), 1–11. https://doi.org/10.1155/2014/432092 doi: 10.1155/2014/432092
|
| [21] |
H. Garg, Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process, Int. J. Intell. Syst., 33 (2018), 1234–1263. https://doi.org/10.1002/int.21979 doi: 10.1002/int.21979
|
| [22] |
M. Lin, J. Wei, Z. Xu, R. Chen, Multiattribute group decision-making based on linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean aggregation operators, Complexity, 2018 (2018), 1–24. https://doi.org/10.1155/2018/9531064 doi: 10.1155/2018/9531064
|
| [23] |
M. Lin, X. Li, L. Chen, Linguistic $q$-rung orthopair fuzzy sets and their interactional partitioned Heronian mean aggregation operators, Int. J. Intell. Syst., 35 (2020), 217–249. https://doi.org/10.1155/2018/9531064 doi: 10.1155/2018/9531064
|
| [24] |
M. Akram, S. Naz, S. A. Edalatpanah, R. Mehreen, Group decision-making framework under linguistic $q$-rung orthopair fuzzy Einstein models, Soft. Comput., 25 (2021), 10309–10334. https://doi.org/10.1007/s00500-021-05771-9 doi: 10.1007/s00500-021-05771-9
|
| [25] |
F. Herrera, L. Martínez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Syst., 8 (2000), 746–752. https://doi.org/10.1109/91.890332 doi: 10.1109/91.890332
|
| [26] |
F. Herrera, L. Martínez, An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making, Int. J. Uncertain. Fuzz. Knowl. Based Syst., 8 (2000), 539–562. https://doi.org/10.1142/S0218488500000381 doi: 10.1142/S0218488500000381
|
| [27] |
X. Liu, H. S. Kim, F. Feng, J. C. R. Alcantud, Centroid transformations of intuitionistic fuzzy values based on aggregation operators, Mathematics, 6 (2018), 215. https://doi.org/10.3390/math6110215 doi: 10.3390/math6110215
|
| [28] |
Y. Liu, Y. Qin, F. Liu, Y. Rong, GIBWM-MABAC approach for MAGDM under multi-granularity intuitionistic 2-tuple linguistic information model, J. Ambient Intell. Hum. Comput., 2021 (2021), 1–17. https://doi.org/10.1007/s12652-021-03476-3 doi: 10.1007/s12652-021-03476-3
|
| [29] |
A. Luqman, M. Akram, J. C. R. Alcantud, Digraph and matrix approach for risk evaluations under Pythagorean fuzzy information, Expert Syst. Appl., 170 (2021), 114518. https://doi.org/10.1016/j.eswa.2020.114518 doi: 10.1016/j.eswa.2020.114518
|
| [30] |
Y. Qin, Y. Liu, S. Abdullah, G. Wei, Group decision support methodology based upon the multigranular generalized orthopair 2-tuple linguistic information model, Int. J. Intell. Syst., 36 (2021), 3367–3400. https://doi.org/10.1002/int.22419 doi: 10.1002/int.22419
|
| [31] |
Y. Xu, H. Wang, Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment, Appl. Soft Comput., 11 (2011), 3988–3997. https://doi.org/10.1016/j.asoc.2011.02.027 doi: 10.1016/j.asoc.2011.02.027
|
| [32] |
M. A. Dulebenets, J. Pasha, M. Kavoosi, O. F. Abioye, E. E. Ozguven, R. Moses, et al., Multiobjective optimization model for emergency evacuation planning in geographical locations with vulnerable population groups, J. Manag. Eng., 36 (2020), 04019043. https://doi.org/10.1061/(ASCE)ME.1943-5479.0000730 doi: 10.1061/(ASCE)ME.1943-5479.0000730
|
| [33] |
J. Pasha, M. A. Dulebenets, A. M. Fathollahi-Fard, G. Tian, Y. Y. Lau, P. Singh, B. Liang, An integrated optimization method for tactical-level planning in liner shipping with heterogeneous ship fleet and environmental considerations, Adv. Eng. Inform., 48 (2021), 101299. https://doi.org/10.1016/j.aei.2021.101299 doi: 10.1016/j.aei.2021.101299
|
| [34] |
J. Pasha, A. L. Nwodu, A. M. Fathollahi-Fard, G. Tian, Z. Li, H. Wang, et al., Exact and metaheuristic algorithms for the vehicle routing problem with a factory-in-a-box in multi-objective settings, Adv. Eng. Inform., 52 (2022), 101623. https://doi.org/10.1016/j.aei.2022.101623 doi: 10.1016/j.aei.2022.101623
|
| [35] |
H. Zhao, C. Zhang, An online-learning-based evolutionary many-objective algorithm, Inf. Sci., 509 (2020), 1–21. https://doi.org/10.1016/j.ins.2019.08.069 doi: 10.1016/j.ins.2019.08.069
|
| [36] |
M. A. Dulebenets, An adaptive polyploid memetic algorithm for scheduling trucks at a cross-docking terminal, Inf. Sci., 565 (2021), 390–421. https://doi.org/10.1016/j.ins.2021.02.039 doi: 10.1016/j.ins.2021.02.039
|
| [37] |
M. Rabbani, N. Oladzad-Abbasabady, N. Akbarian-Saravi, Ambulance routing in disaster response considering variable patient condition: NSGA-Ⅱ and MOPSO algorithms, J. Ind. Manage. Optim., 18 (2022), 1035. https://doi.org/10.3934/jimo.2021007 doi: 10.3934/jimo.2021007
|
| [38] |
T. Y. Chen, Novel generalized distance measure of Pythagorean fuzzy sets and a compromise approach for multiple criteria decision analysis under uncertainty, IEEE Access, 7 (2019), 58168–58185. https://doi.org/10.1109/ACCESS.2019.2914703 doi: 10.1109/ACCESS.2019.2914703
|
| [39] |
H. Garg, D. Rani, Robust averaging-geometric aggregation operators for complex intuitionistic fuzzy sets and their applications to MCDM process, Arab. J. Sci. Eng. 45 (2020), 2017–2033. https://doi.org/10.1007/s13369-019-03925-4 doi: 10.1007/s13369-019-03925-4
|
| [40] |
C. Li, S. Zeng, T. Pan, L. Zheng, A method based on induced aggregation operators and distance measures to multiple attribute decision making under 2-tuple linguistic environment, J. Comput. Syst. Sci., 80 (2014), 1339–1349. https://doi.org/10.1016/j.jcss.2014.03.004 doi: 10.1016/j.jcss.2014.03.004
|
| [41] |
J. H. Park, J. M. Park, Y. C. Kwun, 2-tuple linguistic harmonic operators and their applications in group decision making, Knowl. Based Syst., 44 (2013), 10–19. https://doi.org/10.1016/j.knosys.2013.01.006 doi: 10.1016/j.knosys.2013.01.006
|
| [42] |
G. W. Wei, Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making, Comput. Ind. Eng., 61 (2011), 32–38. https://doi.org/10.1016/j.cie.2011.02.007 doi: 10.1016/j.cie.2011.02.007
|
| [43] |
F. Zhou, T. Y. Chen, An integrated multicriteria group decision-making approach for green supplier selection under Pythagorean fuzzy scenarios, IEEE Access, 8 (2020), 165216–165231. https://doi.org/10.1109/ACCESS.2020.3022377 doi: 10.1109/ACCESS.2020.3022377
|
| [44] |
P. Liu, S. Naz, M. Akram, M. Muzammal, Group decision-making analysis based on linguistic $q$-rung orthopair fuzzy generalized point weighted aggregation operators, Int. J. Mach. Learn. Cybern., 13 (2021), 883–906. https://doi.org/10.1007/s13042-021-01425-2 doi: 10.1007/s13042-021-01425-2
|
| [45] | H. Hamacher, Über logische verknünpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen, in Progress in Cybernetics and Systems Research (Eds. Trappl, Klir, Riccardi), Hemisphere, Washington DC, 3 (1978), 276–288. |
| [46] |
S. Faizi, W. Salabun, S. Nawaz, A. U. Rehman, J. W
|
| [47] |
S. Abdullah, O. Barukab, M. Qiyas, M. Arif, S. A. Khan, Analysis of decision support system based on 2-tuple spherical fuzzy linguistic aggregation information, Appl. Sci., 10 (2020), 276. https://doi.org/10.3390/app10010276 doi: 10.3390/app10010276
|
| [48] |
M. Akram, G. Shahzadi, J. C. R. Alcantud, Multi-attribute decision-making with q-rung picture fuzzy information, Granul. Comput., 7 (2022), 197–215. https://doi.org/10.1007/s41066-021-00260-8 doi: 10.1007/s41066-021-00260-8
|
| [49] |
X. P. Jiang, G. W. Wei, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, J. Intell. Fuzzy Syst., 27 (2014), 2153–2162. https://doi.org/10.3233/IFS-141180 doi: 10.3233/IFS-141180
|
| [50] |
N. Waseem, M. Akram, J. C. R. Alcantud, Multi-attribute decision-making based on $m$-polar fuzzy Hamacher aggregation operators, Symmetry, 11 (2019), 1498. https://doi.org/10.3390/sym11121498 doi: 10.3390/sym11121498
|
| [51] |
Z. Zhang, F. Wei, S. Zhou, Approaches to comprehensive evaluation with 2-tuple linguistic information, J. Intell. Fuzzy Syst., 28 (2015), 469–475. https://doi.org/10.3233/IFS-141323 doi: 10.3233/IFS-141323
|
| [52] |
H. Jin, S. Ashraf, S. Abdullah, M. Qiyas, M. Bano, S. Zeng, Linguistic spherical fuzzy aggregation operators and their applications in multi-attribute decision making problems, Mathematics, 7 (2019), 413. https://doi.org/10.3390/math7050413 doi: 10.3390/math7050413
|
| [53] |
Y. Ju, A. Wang, J. Ma, H. Gao, E. D. Santibanez Gonzalez, Some $q$-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making, Int. J. Intell. Syst., 35 (2020), 184–213. https://doi.org/10.1002/int.22205 doi: 10.1002/int.22205
|
| [54] |
X. Deng, J. Wang, G. Wei, Some 2-tuple linguistic Pythagorean Heronian mean operators and their application to multiple attribute decision-making, J. Exp. Theor. Artif. Intell., 31 (2019), 555–574. https://doi.org/10.1080/0952813X.2019.1579258 doi: 10.1080/0952813X.2019.1579258
|
| [55] |
X. Deng, J. Wang, G. Wei, M. Lu, Models for multiple attribute decision making with some 2-tuple linguistic Pythagorean fuzzy Hamy mean operators, Mathematics, 6 (2018), 236. https://doi.org/10.3390/math6110236 doi: 10.3390/math6110236
|
| [56] |
G. W. Wei, 2-tuple intuitionistic fuzzy linguistic aggregation operators in multiple attribute decision making, Iran. J. Fuzzy Syst., 16 (2019), 159–174. https://doi.org/10.22111/IJFS.2019.4789 doi: 10.22111/IJFS.2019.4789
|
| [57] |
M. Lu, G. Wei, F. E. Alsaadi, T. Hayat, A. Alsaedi, Bipolar 2-tuple linguistic aggregation operators in multiple attribute decision making, J. Intell. Fuzzy Syst., 33 (2017), 1197–1207. https://doi.org/10.3233/JIFS-16946 doi: 10.3233/JIFS-16946
|
| [58] |
Y. Zhang, G. Wei, Y. Guo, C. Wei, TODIM method based on cumulative prospect theory for multiple attribute group decision-making under 2-tuple linguistic Pythagorean fuzzy environment, Int. J. Intell. Syst., 36 (2021), 2548–2571. https://doi.org/10.1002/int.22393 doi: 10.1002/int.22393
|
| [59] |
M. Akram, R. Bibi, M. A. Al-Shamiri, A decision-making framework based on 2-tuple linguistic Fermatean fuzzy Hamy mean operators, Math. Probl. Eng., 2022 (2022), 1501880. https://doi.org/10.1155/2022/1501880 doi: 10.1155/2022/1501880
|
| [60] |
M. Akram, A. Khan, U. Ahmad, Extended MULTIMOORA method based on 2-tuple linguistic Pythagorean fuzzy sets for multi-attribute group decision-making, Granul. Comput., 2022 (2022), 1–22. https://doi.org/10.1007/s41066-022-00330-5. doi: 10.1007/s41066-022-00330-5
|
| [61] |
D. Ramot, M. Friedman, G. Langholz, G. A. Kandel, Complex fuzzy logic, IEEE Trans. Fuzzy Syst., 11 (2003), 450–461. https://doi.org/10.1109/TFUZZ.2003.814832 doi: 10.1109/TFUZZ.2003.814832
|
| [62] |
D. Ramot, R. Milo, M. Fiedman, A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst., 10 (2002), 171–186. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
|
| [63] | A. M. D. J. S. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, in AIP Conference Proceedings, 1482 (2012), 464–470. https://doi.org/10.1063/1.4757515 |
| [64] |
A. U. M. Alkouri, A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstr. Appl. Anal., 2013 (2013), 1–18. https://doi.org/10.1155/2013/287382 doi: 10.1155/2013/287382
|
| [65] |
Y. Rong, Y. Liu, Z. Pei, Complex $q$-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean operators and its application to emergency program selection, Int. J. Intell. Syst., 35 (2020), 1749–1790. https://doi.org/10.1002/int.22271 doi: 10.1002/int.22271
|
| [66] |
L. Bi, S. Dai, B. Hu, Complex fuzzy geometric aggregation operators, Symmetry, 10 (2018), 251, https://doi.org/10.3390/sym10070251. doi: 10.3390/sym10070251
|
| [67] |
L. Bi, S. Dai, B. Hu, S. Li, Complex fuzzy arithmetic aggregation operators, J. Intell. Fuzzy Syst., 36 (2019), 2765–2771. https://doi.org/10.3233/JIFS-18568 doi: 10.3233/JIFS-18568
|
| [68] |
P. Liu, T. Mahmood, Z. Ali, Complex $q$-rung orthopair fuzzy aggregation operators and their applications in multiattribute group decision making, Information, 11 (2020), 5. https://doi.org/10.3390/info11010005 doi: 10.3390/info11010005
|
| [69] |
A. Luqman, M. Akram, A. N. Al-Kenani, J. C. R. Alcantud, A study on hypergraph representations of complex fuzzy information, Symmetry, 11 (2019), 1381. https://doi.org/10.3390/sym11111381 doi: 10.3390/sym11111381
|
| [70] |
S. Naz, M. Akram, M. M. A. Al-Shamiri, M. M. Khalaf, G. Yousaf, A new MAGDM method with 2-tuple linguistic bipolar fuzzy Heronian mean operators, Math. Biosci. Eng., 19 (2022), 3843–3878. https://doi.org/10.3934/mbe.2022177 doi: 10.3934/mbe.2022177
|
| [71] |
S. Naz, M. Akram, M. M. A. Al-Shamiri, M. R. Saeed, Evaluation of network security service provider using 2-tuple linguistic complex$q$-rung orthopair fuzzy COPRAS method, Complexity, 2022 (2022), 1–27. https://doi.org/10.1155/2022/4523287 doi: 10.1155/2022/4523287
|
| [72] |
P. Liu, Z. Ali, T. Mahmood, Generalized complex $q$-rung orthopair fuzzy Einstein averaging aggregation operators and their application in multi-attribute decision making, Complex Intell. Syst., 7 (2021), 511–538. https://doi.org/10.1007/s40747-020-00197-6 doi: 10.1007/s40747-020-00197-6
|
| [73] |
M. Akram, A. Bashir, S. A. Edalatpanah, A hybrid decision-making analysis under complex $q$-rung picture fuzzy Einstein averaging operators, Comput. Appl. Math., 40 (2021), 1–35. https://doi.org/10.1007/s40314-021-01651-y doi: 10.1007/s40314-021-01651-y
|
| [74] |
M. Akram, X. Peng, A. Sattar, Multi-criteria decision-making model using complex Pythagorean fuzzy Yager aggregation operators, Arab. J. Sci. Eng., 46 (2021), 1691–1717. https://doi.org/10.1007/s13369-020-04864-1 doi: 10.1007/s13369-020-04864-1
|
| [75] |
H. Garg, D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arab. J. Sci. Eng., 44 (2019), 2679–2698. https://doi.org/10.1007/s13369-018-3413-x doi: 10.1007/s13369-018-3413-x
|
| [76] |
P. Liu, Z. Ali, T. Mahmood, Novel complex $T$-spherical fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their application to multi-attribute decision-making, Int. J. Comput. Intell. Syst., 14 (2021), 295–331. https://doi.org/10.2991/ijcis.d.201207.003 doi: 10.2991/ijcis.d.201207.003
|
| [77] |
M. Akram, S. Naz, F. Feng, A. Shafiq, Assessment of hydropower plants in Pakistan: Muirhead mean-based 2-tuple linguistic $T$-spherical fuzzy model combining SWARA with COPRAS, Arabian J. Sci. Eng., 2022 (2022), 1–30. https://doi.org/10.1007/s13369-022-07081-0 doi: 10.1007/s13369-022-07081-0
|
| [78] |
M. Akram, N. Ramzan, F. Feng, Extending COPRAS method with linguistic Fermatean fuzzy sets and Hamy mean operators, J. Math., 2022 (2022), 8239263. https://doi.org/10.1155/2022/8239263 doi: 10.1155/2022/8239263
|
| [79] |
T. Mahmood, Z. Ali, A novel approach of complex $q$-rung orthopair fuzzy Hamacher aggregation operators and their application for cleaner production assessment in gold mines, J. Ambient Intell. Humaniz. Comput., 12 (2021), 8933–8959. https://doi.org/10.1007/s12652-020-02697-2 doi: 10.1007/s12652-020-02697-2
|
| [80] |
D. Rani, H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making, Expert Syst., 35 (2018), e12325. https://doi.org/10.1111/exsy.12325 doi: 10.1111/exsy.12325
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Muhammad Akram, Ayesha Khan, Uzma Ahmad, José Carlos R. Alcantud, Mohammed M. Ali Al-Shamiri. A new group decision-making framework based on 2-tuple linguistic complex $ q $-rung picture fuzzy sets[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11281-11323. doi: 10.3934/mbe.2022526
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