Complex interval-value intuitionistic fuzzy sets: Quaternion number representation, correlation coefficient and applications

Yanhong Su; Zengtai Gong; Na Qin; Yanhong Su; Zengtai Gong; Na Qin

doi:10.3934/math.2024973

AIMS Mathematics

2024, Volume 9, Issue 8: 19943-19966. doi: 10.3934/math.2024973

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Research article

Complex interval-value intuitionistic fuzzy sets: Quaternion number representation, correlation coefficient and applications

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2.
College of Computer Science and Engineering, Northwest Normal University, Lanzhou 730070, China

Received: 23 April 2024 Revised: 04 June 2024 Accepted: 12 June 2024 Published: 19 June 2024
MSC : 08A72, 26E50, 03E72

Complex interval-valued intuitionistic fuzzy sets not only consider uncertainty and periodicity semantics at the same time but also choose to express the information value with an interval value to give experts more freedom and make the solution to the problem more reasonable. In this study, we used the interval quaternion number space to generalize and extend the utility of complex interval-valued intuitionistic fuzzy sets, analyze their order relation, and offer new operations based on interval quaternion numbers. We proposed a new score function and correlation coefficient under interval quaternion representation. We applied the interval quaternion representation and correlation coefficient to a multi-criterion decision making model and applied the model to enterprise decision-making.
- complex interval-valued intuitionistic fuzzy sets,
- interval quaternion number,
- score function,
- correlation coefficient,
- multi-criteria decision making
Citation: Yanhong Su, Zengtai Gong, Na Qin. Complex interval-value intuitionistic fuzzy sets: Quaternion number representation, correlation coefficient and applications[J]. AIMS Mathematics, 2024, 9(8): 19943-19966. doi: 10.3934/math.2024973

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Abstract

Complex interval-valued intuitionistic fuzzy sets not only consider uncertainty and periodicity semantics at the same time but also choose to express the information value with an interval value to give experts more freedom and make the solution to the problem more reasonable. In this study, we used the interval quaternion number space to generalize and extend the utility of complex interval-valued intuitionistic fuzzy sets, analyze their order relation, and offer new operations based on interval quaternion numbers. We proposed a new score function and correlation coefficient under interval quaternion representation. We applied the interval quaternion representation and correlation coefficient to a multi-criterion decision making model and applied the model to enterprise decision-making.

References

[1]	L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[2]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[3]	S. P. Wan, J. Y. Dong, S. M. Chen, A novel intuitionistic fuzzy best-worst method for group decision making with intuitionistic fuzzy preference relations, Inform. Sci., 666 (2024), 120404. https://doi.org/10.1016/j.ins.2024.120404 doi: 10.1016/j.ins.2024.120404
[4]	K. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Set. Syst., 64 (1994), 159–174. https://doi.org/10.1016/0165-0114(94)90331-X doi: 10.1016/0165-0114(94)90331-X
[5]	K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Set. Syst., 31 (1989), 343–349. https://doi.org/10.1016/0165-0114(89)90205-4 doi: 10.1016/0165-0114(89)90205-4
[6]	J. Y. Dong, X. Y. Lu, H. C. Li, S. P. Wan, S. Q. Yang, Consistency and consensus enhancing in group decision making with interval-valued intuitionistic multiplicative preference relations based on bounded confidence, Inform. Sci., 652 (2024), 119727. https://doi.org/10.1016/j.ins.2023.119727 doi: 10.1016/j.ins.2023.119727
[7]	J. Y. Dong, S. P. Wan, Type-2 interval-valued intuitionstic fuzzy matrix game and application to energy vehicle industry development, Expert Syst. Appl., 249 (2024), 123398. https://doi.org/10.1016/j.eswa.2024.123398 doi: 10.1016/j.eswa.2024.123398
[8]	J. Y. Dong, S. P. Wan, Interval-valued intuitionistic fuzzy best-worst method with additive consistency, Expert Syst. Appl., 236 (2024), 121213. https://doi.org/10.1016/j.eswa.2023.121213 doi: 10.1016/j.eswa.2023.121213
[9]	S. M. Chen, S. H. Cheng, W. H. Tsai, Multiple attribute group decision making based on interval-valued intuitionistic fuzzy aggregation operators and transformation techniques of interval-valued intuitionistic fuzzy values, Inform. Sci., 1 (2016), 367–368. https://doi.org/10.1016/j.ins.2016.05.041 doi: 10.1016/j.ins.2016.05.041
[10]	H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Comput. Indust. Eng., 101 (2016), 53–69. https://doi.org/10.1016/j.cie.2016.08.017 doi: 10.1016/j.cie.2016.08.017
[11]	H. Garg, Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Eng. Appl. Artif. Intel., 60 (2017), 164–174. https://doi.org/10.1016/j.engappai.2017.02.008 doi: 10.1016/j.engappai.2017.02.008
[12]	H. Garg, Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process, J. Ind. Manag. Optim., 14 (2018), 283–308. https://doi.org/10.3934/jimo.2017047 doi: 10.3934/jimo.2017047
[13]	P. Liu, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE T. Fuzzy Syst., 22 (2014), 83–97. https://doi.org/10.1109/TFUZZ.2013.2248736 doi: 10.1109/TFUZZ.2013.2248736
[14]	K. Ullah, H. Garg, Z. Gul, T. Mahmood, Q. Khan, Z. Ali, Interval valued T-spherical fuzzy information aggregation based on Dombi t-Norm and Dombi t-Conorm for multi-attribute decision making problems, Symmetry, 13 (2021), 1053. https://doi.org/10.3390/sym13061053 doi: 10.3390/sym13061053
[15]	H. Garg, K. Kumar, A novel possibility measure to interval-valued intuitionistic fuzzy set using connection number of set pair analysis and its applications, Neural Comput. Appl., 32 (2020), 3337–3348. https://doi.org/10.1007/s00521-019-04291-w doi: 10.1007/s00521-019-04291-w
[16]	A. Tiwari, Q. D. Lohani, P. K. Muhuri, Interval-valued intuitionistic fuzzy TOPSIS method for supplier selection problem, In: 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2020, 1–8. https://doi.org/10.1109/FUZZ48607.2020.9177852
[17]	F. Wang, S. Wan, Possibility degree and divergence degree based method for intervalvalued intuitionistic fuzzy multi-attribute group decision making, Expert Syst. Appl., 141 (2020), 112929. https://doi.org/10.1016/j.eswa.2019.112929 doi: 10.1016/j.eswa.2019.112929
[18]	Z. S. Xu, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optim. Decis. Ma., 6 (2007), 109–121. https://doi.org/ 10.1007/s10700-007-9004-z doi: 10.1007/s10700-007-9004-z
[19]	J. Bharatraj, Interval valued intuitionistic fuzzy Gaussian membership function: A novel extension, In: International Conference on Intelligent and Fuzzy Systems, 2021,372–380. https://doi.org/10.1007/978-3-030-51156-2-44
[20]	A. R. Mishra, P. Rani, A. Mardani, K. R. Pardasani, K. Govindan, M. Alrasheedi, Healthcare evaluation in hazardous waste recycling using novel interval-valued intuitionistic fuzzy information based on complex proportional assessment method, Comput. Indust. Eng., 139 (2020), 106140. https://doi.org/10.1016/j.cie.2019.106140 doi: 10.1016/j.cie.2019.106140
[21]	Y. Wang, Y. Shi, Measuring the service quality of urban rail transit based on intervalvalued intuitionistic fuzzy model, KSCE J. Civ. Eng., 24 (2020), 647–656. https://doi.org/10.1007/s12205-020-0937-x doi: 10.1007/s12205-020-0937-x
[22]	D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE T. Fuzzy Syst., 10 (2002), 171–186. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
[23]	Z. T. Gong, F. D. Wang, Complex fuzzy sets:(r, $\theta$)-cut sets, decomposition theorems, extension principles and their applications, J. Intel. Fuzzy Syst., 44 (2023), 8147–8162. https://doi.org/10.3233/JIFS-221639 doi: 10.3233/JIFS-221639
[24]	A. M. D. J. S. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, AIP Conf. Proc., 1482 (2012), 464–470. https://doi.org/10.1063/1.4757515 doi: 10.1063/1.4757515
[25]	A. U. M. Alkouri, A. R. Salleh, Some operations on complex Atanassov's intuitionistic fuzzy sets, AIP Conf. Proc., 1571 (2013), 987–993. https://doi.org/10.1063/1.4858782 doi: 10.1063/1.4858782
[26]	Z. T. Gong, F. D. Wang, Operation properties and ($\alpha$, $\beta$)-equalities of complex intuitionistic fuzzy sets, Soft Comput., 27 (2023), 4369–4391. https://doi.org/10.1007/s00500-023-07854-1 doi: 10.1007/s00500-023-07854-1
[27]	H. Garg, D. Rani, Complex interval-valued intuitionistic fuzzy sets and their aggregation operators, Fund. Inform., 164 (2019), 61–101. https://doi.org/10.3233/FI-2019-1755 doi: 10.3233/FI-2019-1755
[28]	D. E. Tamir, M. Ali, N. D. Rishe, A. Kandel, Complex number representation of intuitionistic fuzzy sets, In: World Conference on Soft Computing, USA, Berkeley, 2016,108–113.
[29]	R. T. Ngan, M. Ali, D. E. Tamir, N. D. Rishe, A. Kandel, Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making, Appl. Soft Comput., 87 (2020), 105961. https://doi.org/10.1016/j.asoc.2019.105961 doi: 10.1016/j.asoc.2019.105961
[30]	L. Pan, Y. Deng, K. H. Cheong, Quaternion model of Pythagorean fuzzy sets and its distance measure, Expert Syst. Appl., 213 (2023), 119222. https://doi.org/10.1016/j.eswa.2022.119222 doi: 10.1016/j.eswa.2022.119222
[31]	R. P. Moura, F. B. Bergamaschi, R. H. Santiago, B. R. Bedregal, Fuzzy quaternion numbers, In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2013, 1–6. https://doi.org/10.1109/FUZZ-IEEE.2013.6622400
[32]	D. Zindani, S. R. Maity, S. Bhowmik, Complex interval-valued intuitionistic fuzzy TODIM approach and its application to group decision making, J. Amb. Intel. Hum. Comput., 12 (2021), 2079–2102. https://doi.org/10.1007/s12652-020-02308-0 doi: 10.1007/s12652-020-02308-0
[33]	M. S. A. Khan, S. U. Jan, R. Jan, T. Senapati, S. Moslem, Complex interval-valued intuitionistic fuzzy decision support system with application to COVID-19 healthcare facilities, Complex Intell. Syst., 9 (2023), 7103–7132. https://doi.org/10.1007/s40747-023-01090-8 doi: 10.1007/s40747-023-01090-8
[34]	G. W. Meng, Basic theory for interval-valued fuzzy sets, Math. Appl., 6 (1993), 212–217.
[35]	M. M. Gao, T. Sun, J. J. Zhu, A new scoring function in multi-criteria decision-making based on Vague set, J. Syst. Sci. Math. Sci., 34 (2014), 96–105.
[36]	D. H. Hong, C. H. Choi, Multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Set. Syst., 114 (2000), 103–113. https://doi.org/10.1016/S0165-0114(98)00271-1 doi: 10.1016/S0165-0114(98)00271-1
[37]	E. Y. Zhang, J. Wang, S. Y. Wang, A new scoring function in multi-criteria decision-making based on Vague set, J. Syst. Sci. Math. Sci., 31 (2011), 961–974.
[38]	T. Gerstenkorn, J. Manko, Correlation of intuitionistic fuzzy sets, Fuzzy Set. Syst., 44 (1991), 39–43. https://doi.org/10.1016/0165-0114(91)90031-K doi: 10.1016/0165-0114(91)90031-K
[39]	H. Bustince, P. Burillo, Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Set. Syst., 74 (1995), 237–244. https://doi.org/10.1016/0165-0114(94)00343-6 doi: 10.1016/0165-0114(94)00343-6
[40]	H. Garg, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, Int. J. Intell. Syst., 31 (2016), 1234–1252. https://doi.org/10.1002/int.21827 doi: 10.1002/int.21827
[41]	H. Garg, Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process, J. Ind. Manag. Optim., 14 (2018), 1501–1519. https://doi.org/10.3934/jimo.2018018 doi: 10.3934/jimo.2018018
[42]	R. Arora, H. Garg, A robust correlation coefficient measure of dual hesitant fuzzy soft sets and their application in decision making, Eng. Appl. Artif. Intel., 72 (2018), 80–92. https://doi.org/10.1016/j.engappai.2018.03.019 doi: 10.1016/j.engappai.2018.03.019
[43]	H. Garg, D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Appl. Intell., 49 (2019), 496–512. https://doi.org/10.1007/s10489-018-1290-3 doi: 10.1007/s10489-018-1290-3
[44]	L. Luo, H. Ren, A new similarity measure of intuitionistic fuzzy set and application in MADM problem, AMSE Ser. Adv. A, 53 (2016), 204–223.
[45]	B. Liu, Y. Shen, L. Mu, X. Chen, L. Chen, A new correlation measure of the intuitionistic fuzzy sets, J. Intel. Fuzzy Syst., 30 (2016), 1019–1028. https://doi.org/10.3233/IFS-151824 doi: 10.3233/IFS-151824
[46]	C. P. Wei, P. Wang, Y. Z. Zhang, Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications, Inform. Sci., 181 (2011), 4273–4286. https://doi.org/10.1016/j.ins.2011.06.001 doi: 10.1016/j.ins.2011.06.001
[47]	H. Garg, D. Rani, Some results on information measures for complex intuitionistic fuzzy sets, Int. J. Intell. Syst., 34 (2019), 2319–2363. https://doi.org/10.1002/int.22127 doi: 10.1002/int.22127
[48]	Z. S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control Decis., 22 (2007), 215–219.
[49]	J. Ye, Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets, Appl. Math. Model., 34 (2010), 3864–3870. https://doi.org/10.1016/j.apm.2010.03.025 doi: 10.1016/j.apm.2010.03.025

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