Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

Harish Garg; Dimple Rani

doi:10.3934/jimo.2020069

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September 2021, 17(5): 2279-2306. doi: 10.3934/jimo.2020069

Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

Harish Garg and Dimple Rani

School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala-147004, Punjab, India

Received August 2019 Revised December 2019 Published September 2021 Early access March 2020

Complex intuitionistic fuzzy sets (CIFSs), characterized by complex-valued grades of membership and non-membership, are a generalization of standard intuitionistic fuzzy (IF) sets that better speak to time-periodic issues and handle two-dimensional data in a solitary set. Under this environment, in this article, various mean-type operators, namely complex IF Bonferroni means (CIFBM) and complex IF weighted Bonferroni mean (CIFWBM) are presented along with their properties and numerous particular cases of CIFBM are discussed. Further, using the presented operators a decision-making approach is developed and is illustrated with the help of a practical example. Also, the reliability of the developed methodology is investigated with the aid of validity test criteria and the example results are compared with prevailing methods based on operators.

Keywords: Complex intuitionistic fuzzy set, Bonferroni mean, geometric operators, multicriteria decision-making, aggregation operators.

Mathematics Subject Classification: Primary: 68T35; 90B50; 62A86; 03E72.

Citation: Harish Garg, Dimple Rani. Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2279-2306. doi: 10.3934/jimo.2020069

References:

[1]	M. Ali and F. Smarandache, Complex neutrosophic set, Neural Computing and Applications, 28 (2017), 1817-1834. Google Scholar
[2]	A. Alkouri and A. Salleh, Complex intuitionistic fuzzy sets, chap. 2nd International Conference on Fundamental and Applied Sciences, 1482 (2012), 464-470. Google Scholar
[3]	A. U. M. Alkouri and A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstract and Applied Analysis, 2013 (2013), Article ID 287382, 18 pages. doi: 10.1155/2013/287382. Google Scholar
[4]	R. Arora and H. Garg, Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties, Computational and Applied Mathematics, 38 (2019), Art. 36, 32 pp. doi: 10.1007/s40314-019-0764-1. Google Scholar
[5]	K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (1989), 343-349. doi: 10.1016/0165-0114(89)90205-4. Google Scholar
[6]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3. Google Scholar
[7]	C. Bonferroni, Sulle medie multiple di potenze, Bollettino Dell'Unione Matematica Italiana, 5 (1950), 267-270. Google Scholar
[8]	S. M. Chen and C. H. Chang, Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Information Sciences, 352/353 (2016), 133-149. doi: 10.1016/j.ins.2016.02.049. Google Scholar
[9]	S. Dick, R. R. Yager and O. Yazdanbakhsh, On Pythagorean and complex fuzzy set operations, IEEE Transactions on Fuzzy Systems, 24 (2016), 1009-1021. Google Scholar
[10]	H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69. doi: 10.1016/j.cie.2016.08.017. Google Scholar
[11]	H. Garg, Some series of intuitionistic fuzzy interactive averaging aggregation operators, SpringerPlus, 5 (2016), 999. doi: 10.1186/s40064-016-2591-9. Google Scholar
[12]	H. Garg, Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Engineering Applications of Artificial Intelligence, 60 (2017), 164-174. doi: 10.1016/j.engappai.2017.02.008. Google Scholar
[13]	H. Garg, Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 43 (2019), 597-613. doi: 10.1007/s40998-018-0167-0. Google Scholar
[14]	H. Garg and R. Arora, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society, 69 (2018), 1711-1724. doi: 10.1080/01605682.2017.1409159. Google Scholar
[15]	H. Garg and K. Kumar, Linguistic interval-valued atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2302-2311. doi: 10.1109/TFUZZ.2019.2897961. Google Scholar
[16]	H. Garg and D. Rani, Complex interval- valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae, 164 (2019), 61-101. doi: 10.3233/FI-2019-1755. Google Scholar
[17]	H. Garg and D. Rani, Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment, Group Decision and Negotiation, 28 (2019), 991-1050. doi: 10.1007/s10726-019-09631-8. Google Scholar
[18]	H. Garg and D. Rani, New generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm, Journal of Experimental and Theoretical Artificial Intelligence, 32 (2020), 81-109. Google Scholar
[19]	H. Garg and D. Rani, Robust Averaging - Geometric aggregation operators for Complex intuitionistic fuzzy sets and their applications to MCDM process, Arabian Journal for Science and Engineering, 45 (2020), 2017-2033. doi: 10.1007/s13369-019-03925-4. Google Scholar
[20]	H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Applied Intelligence, 49 (2019), 496-512. doi: 10.1007/s10489-018-1290-3. Google Scholar
[21]	H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2679-2698. Google Scholar
[22]	H. Garg and D. Rani, Some results on information measures for complex intuitionistic fuzzy sets, International Journal of Intelligent Systems, 34 (2019), 2319-2363. doi: 10.1002/int.22127. Google Scholar
[23]	M. Goyal, D. Yadav and A. Tripathi, Intuitionistic fuzzy genetic weighted averaging operator and its application for multiple attribute decision making in E-learning, Indian Journal of Science and Technology, 9 (2016), 1-15. doi: 10.17485/ijst/2016/v9i1/76191. Google Scholar
[24]	Y. He, H. Chen, L. Zhau, J. Liu and Z. Tao, Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making, Information Sciences, 259 (2014), 142-159. doi: 10.1016/j.ins.2013.08.018. Google Scholar
[25]	J. Y. Huang, Intuitionistic fuzzy Hamacher aggregation operator and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 27 (2014), 505-513. doi: 10.3233/IFS-131019. Google Scholar
[26]	G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427. doi: 10.1615/Int.J.UncertaintyQuantification.2018020471. Google Scholar
[27]	G. Kaur and H. Garg, Generalized cubic intuitionistic fuzzy aggregation operators using t-norm operations and their applications to group decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2775-2794. doi: 10.1615/Int.J.UncertaintyQuantification.2018020471. Google Scholar
[28]	G. Kaur and H. Garg, Multi - attribute decision - making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065. Google Scholar
[29]	K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119. doi: 10.1007/s10489-017-1067-0. Google Scholar
[30]	T. Kumar and R. K. Bajaj, On complex intuitionistic fuzzy soft sets with distance measures and entropies, Journal of Mathematics, 2014 (2014), Article ID 972198, 12 pages. doi: 10.1155/2014/972198. Google Scholar
[31]	D. Li, W. Zeng and J. Li, Geometric bonferroni mean operators, International Journal of Intelligent Systems, 31 (2016), 1181-1197. doi: 10.1002/int.21822. Google Scholar
[32]	C. Maclaurin, A second letter to martin folkes, esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A, 36 (1729), 59-96. Google Scholar
[33]	A. A. Quran and N. Hassan, The complex neutrosophic soft expert set and its application in decision making, Journal of Intelligent & Fuzzy Systems, 34 (2018), 569-582. Google Scholar
[34]	D. Ramot, M. Friedman, G. Langholz and A. Kandel, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems, 11 (2003), 450-461. Google Scholar
[35]	D. Ramot, R. Milo, M. Fiedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 10 (2002), 171-186. Google Scholar
[36]	D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439. doi: 10.1615/Int.J.UncertaintyQuantification.2017020356. Google Scholar
[37]	D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, 35 (2018), e12325. Google Scholar
[38]	R. Verma, Generalized bonferroni mean operator for fuzzy number intuitionistic fuzzy sets and its application to multiattribute decision making, International Journal of Intelligent Systems, 30 (2015), 499-519. doi: 10.1002/int.21705. Google Scholar
[39]	W. Wang and X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy Systems, 20 (2012), 923-938. doi: 10.1109/TFUZZ.2012.2189405. Google Scholar
[40]	X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega - International Journal of Management Science, 36 (2008), 45-63. doi: 10.1016/j.omega.2005.12.003. Google Scholar
[41]	Z. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems, 24 (2011), 749-760. doi: 10.1016/j.knosys.2011.01.011. Google Scholar
[42]	Z. Xu and R. R. Yager, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, 41 (2011), 568-578. Google Scholar
[43]	Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15 (2007), 1179-1187. Google Scholar
[44]	Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433. doi: 10.1080/03081070600574353. Google Scholar
[45]	R. R. Yager, On generalized bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning, 50 (2009), 1279-1286. doi: 10.1016/j.ijar.2009.06.004. Google Scholar
[46]	R. R. Yager and A. M. Abbasov, Pythagorean membeship grades, complex numbers and decision making, International Journal of Intelligent Systems, 28 (2013), 436-452. Google Scholar
[47]	O. Yazdanbakhsh and S. Dick, A systematic review of complex fuzzy sets and logic, Fuzzy Sets and Systems, 338 (2018), 1-22. doi: 10.1016/j.fss.2017.01.010. Google Scholar
[48]	J. Ye, Intuitionistic fuzzy hybrid arithmetic and geometric aggregation operators for the decision-making of mechanical design schemes, Applied Intelligence, 47 (2017), 743-751. doi: 10.1007/s10489-017-0930-3. Google Scholar
[49]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[50]	W. Zhou and Z. Xu, Extreme intuitionistic fuzzy weighted aggregation operators and their applications in optimism and pessimism decision-making processes, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1129-1138. Google Scholar

show all references

References:

[1]	M. Ali and F. Smarandache, Complex neutrosophic set, Neural Computing and Applications, 28 (2017), 1817-1834. Google Scholar
[2]	A. Alkouri and A. Salleh, Complex intuitionistic fuzzy sets, chap. 2nd International Conference on Fundamental and Applied Sciences, 1482 (2012), 464-470. Google Scholar
[3]	A. U. M. Alkouri and A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstract and Applied Analysis, 2013 (2013), Article ID 287382, 18 pages. doi: 10.1155/2013/287382. Google Scholar
[4]	R. Arora and H. Garg, Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties, Computational and Applied Mathematics, 38 (2019), Art. 36, 32 pp. doi: 10.1007/s40314-019-0764-1. Google Scholar
[5]	K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (1989), 343-349. doi: 10.1016/0165-0114(89)90205-4. Google Scholar
[6]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3. Google Scholar
[7]	C. Bonferroni, Sulle medie multiple di potenze, Bollettino Dell'Unione Matematica Italiana, 5 (1950), 267-270. Google Scholar
[8]	S. M. Chen and C. H. Chang, Fuzzy multiattribute decision making based on transformation techniques of intuitionistic fuzzy values and intuitionistic fuzzy geometric averaging operators, Information Sciences, 352/353 (2016), 133-149. doi: 10.1016/j.ins.2016.02.049. Google Scholar
[9]	S. Dick, R. R. Yager and O. Yazdanbakhsh, On Pythagorean and complex fuzzy set operations, IEEE Transactions on Fuzzy Systems, 24 (2016), 1009-1021. Google Scholar
[10]	H. Garg, Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making, Computers and Industrial Engineering, 101 (2016), 53-69. doi: 10.1016/j.cie.2016.08.017. Google Scholar
[11]	H. Garg, Some series of intuitionistic fuzzy interactive averaging aggregation operators, SpringerPlus, 5 (2016), 999. doi: 10.1186/s40064-016-2591-9. Google Scholar
[12]	H. Garg, Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application, Engineering Applications of Artificial Intelligence, 60 (2017), 164-174. doi: 10.1016/j.engappai.2017.02.008. Google Scholar
[13]	H. Garg, Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 43 (2019), 597-613. doi: 10.1007/s40998-018-0167-0. Google Scholar
[14]	H. Garg and R. Arora, Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society, 69 (2018), 1711-1724. doi: 10.1080/01605682.2017.1409159. Google Scholar
[15]	H. Garg and K. Kumar, Linguistic interval-valued atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2302-2311. doi: 10.1109/TFUZZ.2019.2897961. Google Scholar
[16]	H. Garg and D. Rani, Complex interval- valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae, 164 (2019), 61-101. doi: 10.3233/FI-2019-1755. Google Scholar
[17]	H. Garg and D. Rani, Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment, Group Decision and Negotiation, 28 (2019), 991-1050. doi: 10.1007/s10726-019-09631-8. Google Scholar
[18]	H. Garg and D. Rani, New generalized Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm, Journal of Experimental and Theoretical Artificial Intelligence, 32 (2020), 81-109. Google Scholar
[19]	H. Garg and D. Rani, Robust Averaging - Geometric aggregation operators for Complex intuitionistic fuzzy sets and their applications to MCDM process, Arabian Journal for Science and Engineering, 45 (2020), 2017-2033. doi: 10.1007/s13369-019-03925-4. Google Scholar
[20]	H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Applied Intelligence, 49 (2019), 496-512. doi: 10.1007/s10489-018-1290-3. Google Scholar
[21]	H. Garg and D. Rani, Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2679-2698. Google Scholar
[22]	H. Garg and D. Rani, Some results on information measures for complex intuitionistic fuzzy sets, International Journal of Intelligent Systems, 34 (2019), 2319-2363. doi: 10.1002/int.22127. Google Scholar
[23]	M. Goyal, D. Yadav and A. Tripathi, Intuitionistic fuzzy genetic weighted averaging operator and its application for multiple attribute decision making in E-learning, Indian Journal of Science and Technology, 9 (2016), 1-15. doi: 10.17485/ijst/2016/v9i1/76191. Google Scholar
[24]	Y. He, H. Chen, L. Zhau, J. Liu and Z. Tao, Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making, Information Sciences, 259 (2014), 142-159. doi: 10.1016/j.ins.2013.08.018. Google Scholar
[25]	J. Y. Huang, Intuitionistic fuzzy Hamacher aggregation operator and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems, 27 (2014), 505-513. doi: 10.3233/IFS-131019. Google Scholar
[26]	G. Kaur and H. Garg, Cubic intuitionistic fuzzy aggregation operators, International Journal for Uncertainty Quantification, 8 (2018), 405-427. doi: 10.1615/Int.J.UncertaintyQuantification.2018020471. Google Scholar
[27]	G. Kaur and H. Garg, Generalized cubic intuitionistic fuzzy aggregation operators using t-norm operations and their applications to group decision-making process, Arabian Journal for Science and Engineering, 44 (2019), 2775-2794. doi: 10.1615/Int.J.UncertaintyQuantification.2018020471. Google Scholar
[28]	G. Kaur and H. Garg, Multi - attribute decision - making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy, 20 (2018), Paper No. 65, 26 pp. doi: 10.3390/e20010065. Google Scholar
[29]	K. Kumar and H. Garg, Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making, Applied Intelligence, 48 (2018), 2112-2119. doi: 10.1007/s10489-017-1067-0. Google Scholar
[30]	T. Kumar and R. K. Bajaj, On complex intuitionistic fuzzy soft sets with distance measures and entropies, Journal of Mathematics, 2014 (2014), Article ID 972198, 12 pages. doi: 10.1155/2014/972198. Google Scholar
[31]	D. Li, W. Zeng and J. Li, Geometric bonferroni mean operators, International Journal of Intelligent Systems, 31 (2016), 1181-1197. doi: 10.1002/int.21822. Google Scholar
[32]	C. Maclaurin, A second letter to martin folkes, esq.; concerning the roots of equations, with demonstration of other rules of algebra, Philos Trans Roy Soc London Ser A, 36 (1729), 59-96. Google Scholar
[33]	A. A. Quran and N. Hassan, The complex neutrosophic soft expert set and its application in decision making, Journal of Intelligent & Fuzzy Systems, 34 (2018), 569-582. Google Scholar
[34]	D. Ramot, M. Friedman, G. Langholz and A. Kandel, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems, 11 (2003), 450-461. Google Scholar
[35]	D. Ramot, R. Milo, M. Fiedman and A. Kandel, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 10 (2002), 171-186. Google Scholar
[36]	D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision - making process, International Journal for Uncertainty Quantification, 7 (2017), 423-439. doi: 10.1615/Int.J.UncertaintyQuantification.2017020356. Google Scholar
[37]	D. Rani and H. Garg, Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making, Expert Systems, 35 (2018), e12325. Google Scholar
[38]	R. Verma, Generalized bonferroni mean operator for fuzzy number intuitionistic fuzzy sets and its application to multiattribute decision making, International Journal of Intelligent Systems, 30 (2015), 499-519. doi: 10.1002/int.21705. Google Scholar
[39]	W. Wang and X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy Systems, 20 (2012), 923-938. doi: 10.1109/TFUZZ.2012.2189405. Google Scholar
[40]	X. Wang and E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega - International Journal of Management Science, 36 (2008), 45-63. doi: 10.1016/j.omega.2005.12.003. Google Scholar
[41]	Z. Xu, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems, 24 (2011), 749-760. doi: 10.1016/j.knosys.2011.01.011. Google Scholar
[42]	Z. Xu and R. R. Yager, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, 41 (2011), 568-578. Google Scholar
[43]	Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15 (2007), 1179-1187. Google Scholar
[44]	Z. S. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35 (2006), 417-433. doi: 10.1080/03081070600574353. Google Scholar
[45]	R. R. Yager, On generalized bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning, 50 (2009), 1279-1286. doi: 10.1016/j.ijar.2009.06.004. Google Scholar
[46]	R. R. Yager and A. M. Abbasov, Pythagorean membeship grades, complex numbers and decision making, International Journal of Intelligent Systems, 28 (2013), 436-452. Google Scholar
[47]	O. Yazdanbakhsh and S. Dick, A systematic review of complex fuzzy sets and logic, Fuzzy Sets and Systems, 338 (2018), 1-22. doi: 10.1016/j.fss.2017.01.010. Google Scholar
[48]	J. Ye, Intuitionistic fuzzy hybrid arithmetic and geometric aggregation operators for the decision-making of mechanical design schemes, Applied Intelligence, 47 (2017), 743-751. doi: 10.1007/s10489-017-0930-3. Google Scholar
[49]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[50]	W. Zhou and Z. Xu, Extreme intuitionistic fuzzy weighted aggregation operators and their applications in optimism and pessimism decision-making processes, Journal of Intelligent and Fuzzy Systems, 32 (2017), 1129-1138. Google Scholar

Figure 1. Variation in score values with parameter $ p $ by fixing $ q $

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Figure 2. Score values of alternatives $ \mathcal{H}_u $ for different values of $ p $, $ q $

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Table 1. Comparison of CIFS model with existing models in literature

Model	Uncertainty	Falsity	Hesitation	Periodicity	Ability to represent
					two-dimensional information
Fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \times $	$ \times $
Interval-valued fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \times $	$ \times $
Intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $	$ \times $
Interval-valued intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $	$ \times $
Complex fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \checkmark $	$ \checkmark $
Interval-valued complex fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \checkmark $	$ \checkmark $
Complex intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $

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Model	Uncertainty	Falsity	Hesitation	Periodicity	Ability to represent
					two-dimensional information
Fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \times $	$ \times $
Interval-valued fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \times $	$ \times $
Intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $	$ \times $
Interval-valued intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $	$ \times $
Complex fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \checkmark $	$ \checkmark $
Interval-valued complex fuzzy set	$ \checkmark $	$ \times $	$ \times $	$ \checkmark $	$ \checkmark $
Complex intuitionistic fuzzy set	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $

Table 2. Input information in the form of the complex intuitionistic fuzzy decision-matrix

	$ \mathcal{C}_1 $	$ \mathcal{C}_2 $	$ \mathcal{C}_3 $	$ \mathcal{C}_4 $
$ \mathcal{H}_1 $	$ \big ( (0.7, 0.9), (0.1, 0.1) \big ) $	$ \big ( (0.8, 0.5), (0.1, 0.4) \big ) $	$ \big ( (0.6, 0.6), (0.3, 0.2) \big ) $	$ \big ( (0.7, 0.7), (0.3, 0.2) \big ) $
$ \mathcal{H}_2 $	$ \big ( (0.7, 0.6), (0.3, 0.3) \big ) $	$ \big ( (0.4, 0.9), (0.2, 0.1) \big ) $	$ \big ( (0.7, 0.7), (0.2, 0.3) \big ) $	$ \big ( (0.4, 0.6), (0.3, 0.1) \big ) $
$ \mathcal{H}_3 $	$ \big ( (0.3, 0.4), (0.6, 0.4) \big ) $	$ \big ( (0.6, 0.6), (0.3, 0.4) \big ) $	$ \big ( (0.3, 0.4), (0.5, 0.6) \big ) $	$ \big ( (0.7, 0.7), (0.1, 0.1) \big ) $
$ \mathcal{H}_4 $	$ \big ( (0.4, 0.8), (0.5, 0.1) \big ) $	$ \big ( (0.7, 0.3), (0.3, 0.3) \big ) $	$ \big ( (0.6, 0.5), (0.1, 0.3) \big ) $	$ \big ( (0.5, 0.5), (0.3, 0.4) \big ) $
$ \mathcal{H}_5 $	$ \big ( (0.9, 0.7), (0.1, 0.2) \big ) $	$ \big ( (0.7, 0.7), (0.2, 0.1) \big ) $	$ \big ( (0.7, 0.6), (0.2, 0.2) \big ) $	$ \big ( (0.8, 0.8), (0.1, 0.1) \big ) $

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	$ \mathcal{C}_1 $	$ \mathcal{C}_2 $	$ \mathcal{C}_3 $	$ \mathcal{C}_4 $
$ \mathcal{H}_1 $	$ \big ( (0.7, 0.9), (0.1, 0.1) \big ) $	$ \big ( (0.8, 0.5), (0.1, 0.4) \big ) $	$ \big ( (0.6, 0.6), (0.3, 0.2) \big ) $	$ \big ( (0.7, 0.7), (0.3, 0.2) \big ) $
$ \mathcal{H}_2 $	$ \big ( (0.7, 0.6), (0.3, 0.3) \big ) $	$ \big ( (0.4, 0.9), (0.2, 0.1) \big ) $	$ \big ( (0.7, 0.7), (0.2, 0.3) \big ) $	$ \big ( (0.4, 0.6), (0.3, 0.1) \big ) $
$ \mathcal{H}_3 $	$ \big ( (0.3, 0.4), (0.6, 0.4) \big ) $	$ \big ( (0.6, 0.6), (0.3, 0.4) \big ) $	$ \big ( (0.3, 0.4), (0.5, 0.6) \big ) $	$ \big ( (0.7, 0.7), (0.1, 0.1) \big ) $
$ \mathcal{H}_4 $	$ \big ( (0.4, 0.8), (0.5, 0.1) \big ) $	$ \big ( (0.7, 0.3), (0.3, 0.3) \big ) $	$ \big ( (0.6, 0.5), (0.1, 0.3) \big ) $	$ \big ( (0.5, 0.5), (0.3, 0.4) \big ) $
$ \mathcal{H}_5 $	$ \big ( (0.9, 0.7), (0.1, 0.2) \big ) $	$ \big ( (0.7, 0.7), (0.2, 0.1) \big ) $	$ \big ( (0.7, 0.6), (0.2, 0.2) \big ) $	$ \big ( (0.8, 0.8), (0.1, 0.1) \big ) $

Table 3. Ranking on changing values of $ p $ and $ q $

Values of $ p $ and $ q $	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
$ p=1 $; $ q=1 $	-0.8301	-0.9395	-1.2343	-1.1190	-0.6852	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=1 $; $ q=2 $	-0.7549	-0.9000	-1.1946	-1.0670	-0.6387	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=2 $	-0.7566	-0.8957	-1.1800	-1.0779	-0.6358	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=3 $	-0.7017	-0.8657	-1.1530	-1.0419	-0.6009	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=3.5 $; $ q=0.1 $	-0.4369	-0.7490	-1.0863	-0.8261	-0.4493	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=4 $; $ q=0.1 $	-0.3898	-0.7234	-1.0694	-0.7911	-0.4178	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=5 $; $ q=0.5 $	-0.3927	-0.7177	-1.0684	-0.7962	-0.4108	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=6 $; $ q=1 $	-0.3966	-0.7141	-1.0639	-0.8049	-0.4050	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $

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Values of $ p $ and $ q $	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
$ p=1 $; $ q=1 $	-0.8301	-0.9395	-1.2343	-1.1190	-0.6852	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=1 $; $ q=2 $	-0.7549	-0.9000	-1.1946	-1.0670	-0.6387	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=2 $	-0.7566	-0.8957	-1.1800	-1.0779	-0.6358	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=2 $; $ q=3 $	-0.7017	-0.8657	-1.1530	-1.0419	-0.6009	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=3.5 $; $ q=0.1 $	-0.4369	-0.7490	-1.0863	-0.8261	-0.4493	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=4 $; $ q=0.1 $	-0.3898	-0.7234	-1.0694	-0.7911	-0.4178	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=5 $; $ q=0.5 $	-0.3927	-0.7177	-1.0684	-0.7962	-0.4108	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
$ p=6 $; $ q=1 $	-0.3966	-0.7141	-1.0639	-0.8049	-0.4050	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $

Table 4. Analysis of the Figures 1(a), 1(b), 1(c) and 1(d)

	Value of $ \wp $	Accuracy for $ p=\wp $	Ranking of the alternatives
			When $ p<\wp $	When $ p=\wp $	When $ p<\wp $
Figure 1(a)	$ 5.55 $	$ H(\mathcal{H}_1)=1.8380 $, $ H(\mathcal{H}_5)=1.8689 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(b)	$ - $	$ - $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(c)	$ 1.593 $	$ H(\mathcal{H}_1)=1.8307 $, $ H(\mathcal{H}_5)=1.8686 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(d)	$ 2.93 $	$ H(\mathcal{H}_1)=1.8191 $, $ H(\mathcal{H}_5)=1.8656 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $

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	Value of $ \wp $	Accuracy for $ p=\wp $	Ranking of the alternatives
			When $ p<\wp $	When $ p=\wp $	When $ p<\wp $
Figure 1(a)	$ 5.55 $	$ H(\mathcal{H}_1)=1.8380 $, $ H(\mathcal{H}_5)=1.8689 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(b)	$ - $	$ - $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(c)	$ 1.593 $	$ H(\mathcal{H}_1)=1.8307 $, $ H(\mathcal{H}_5)=1.8686 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $
Figure 1(d)	$ 2.93 $	$ H(\mathcal{H}_1)=1.8191 $, $ H(\mathcal{H}_5)=1.8656 $	$ \mathcal{H}_1\succ \mathcal{H}_5\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $	$ \mathcal{H}_5\succ \mathcal{H}_1\succ \mathcal{H}_2\succ \mathcal{H}_4\succ \mathcal{H}_3 $

Table 5. Comparative Analysis results with CIFS studies

Method used	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
Method based on CIFWA operator [19]	1.1605	0.8812	0.3491	0.6484	1.2545	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on CIFWPA operator [37]	1.1449	0.8829	0.3540	0.6432	1.2504	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Distance measure [3]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Euclidean distance measure [36]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Correlation coefficient [20]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=1) $	-0.8301	-0.9395	-1.2343	-1.1190	-0.6852	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=10) $	-0.2174	-0.6143	-0.9993	-0.6744	-0.2686	$ \mathcal{H}_1 \succ \mathcal{H}_5 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=0) $	-0.6921	-0.8863	-1.1942	-1.0262	-0.6041	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
$ Used: t(a)=-\log(a) $ for $ 0< a \leq 1 $ with $ t(0)=\infty $ in [19], $ \alpha_1=\beta_1=\sigma_1=\alpha_2=\beta_2=\sigma_2=\frac{1}{3} $ in [3]

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Method used	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
Method based on CIFWA operator [19]	1.1605	0.8812	0.3491	0.6484	1.2545	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on CIFWPA operator [37]	1.1449	0.8829	0.3540	0.6432	1.2504	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Distance measure [3]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Euclidean distance measure [36]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Method based on Correlation coefficient [20]	-					$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=1) $	-0.8301	-0.9395	-1.2343	-1.1190	-0.6852	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=10) $	-0.2174	-0.6143	-0.9993	-0.6744	-0.2686	$ \mathcal{H}_1 \succ \mathcal{H}_5 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Proposed method with $ (p=1;q=0) $	-0.6921	-0.8863	-1.1942	-1.0262	-0.6041	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
$ Used: t(a)=-\log(a) $ for $ 0< a \leq 1 $ with $ t(0)=\infty $ in [19], $ \alpha_1=\beta_1=\sigma_1=\alpha_2=\beta_2=\sigma_2=\frac{1}{3} $ in [3]

Table 6. Comparative Analysis results with IFS studies

Method used	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
Xu and Yager [42] method based on IFWBM operator $ (p=1;q=1) $	-0.3968	-0.5370	-0.6319	-0.5754	-0.3136	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [41] method based on IFPWA operator	0.5653	0.3332	0.1484	0.2441	0.6839	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Wang and Liu [39] method based on IFEWA operator	0.5670	0.3276	0.1183	0.2181	0.6871	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu and Yager [44] based on IFWG operator	0.5314	0.2826	-0.0179	0.1466	0.6536	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [43] method based on IFWA operator	0.5701	0.3351	0.1432	0.2301	0.6898	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [10] method based on IFEWGIA operator	0.6563	0.4787	0.0142	0.2849	0.7193	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
He et al. [24] method based on IFGIA method	0.6484	0.4768	-0.0085	0.2707	0.7172	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Huang [25] method based on IFHWA operator	0.5658	0.3241	0.1064	0.2127	0.6860	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [11] method	0.4307	0.1603	0.0710	0.0694	0.6375	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Chen and Chang [8] method	0.4339	0.1804	0.1000	0.0845	0.6435	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Goyal et al.[23] method	0.7982	0.6623	0.3109	0.4510	0.8604	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [12] method	0.4316	0.1669	0.0809	0.0743	0.6392	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Ye [48] method	0.5506	0.3084	0.0596	0.1876	0.6715	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Zhou and Xu [50] method	0.5868	0.3824	0.3288	0.3776	0.6979	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=1) $	-1.3968	-1.5370	-1.6319	-1.5754	-1.3136	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=0) $	-1.3485	-1.5108	-1.6119	-1.5700	-1.2525	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Abbreviations. IFWA: Intuitionistic fuzzy weighted averaging; IFWG: Intuitionistic fuzzy weighted geometric; IFEWA: intuitionistic fuzzy Einstein weighted averaging; IFPWA: intuitionistic fuzzy power weighted averaging; IFWBM: intuitionistic fuzzy weighted Bonferroni mean; IFGIA: intuitionistic fuzzy geometric interactive averaging; IFEWGIA: intuitionistic fuzzy Einstein weighted geometric interactive averaging; IFHWA: intuitionistic fuzzy Hamacher weighted averaging; CIFWBM: complex intuitionistic fuzzy weighted Bonferroni mean.

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Method used	Score values					Ranking
	$ \mathcal{H}_1 $	$ \mathcal{H}_2 $	$ \mathcal{H}_3 $	$ \mathcal{H}_4 $	$ \mathcal{H}_5 $
Xu and Yager [42] method based on IFWBM operator $ (p=1;q=1) $	-0.3968	-0.5370	-0.6319	-0.5754	-0.3136	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [41] method based on IFPWA operator	0.5653	0.3332	0.1484	0.2441	0.6839	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Wang and Liu [39] method based on IFEWA operator	0.5670	0.3276	0.1183	0.2181	0.6871	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu and Yager [44] based on IFWG operator	0.5314	0.2826	-0.0179	0.1466	0.6536	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Xu [43] method based on IFWA operator	0.5701	0.3351	0.1432	0.2301	0.6898	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [10] method based on IFEWGIA operator	0.6563	0.4787	0.0142	0.2849	0.7193	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
He et al. [24] method based on IFGIA method	0.6484	0.4768	-0.0085	0.2707	0.7172	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Huang [25] method based on IFHWA operator	0.5658	0.3241	0.1064	0.2127	0.6860	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [11] method	0.4307	0.1603	0.0710	0.0694	0.6375	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Chen and Chang [8] method	0.4339	0.1804	0.1000	0.0845	0.6435	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Goyal et al.[23] method	0.7982	0.6623	0.3109	0.4510	0.8604	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Garg [12] method	0.4316	0.1669	0.0809	0.0743	0.6392	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_3 \succ \mathcal{H}_4 $
Ye [48] method	0.5506	0.3084	0.0596	0.1876	0.6715	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Zhou and Xu [50] method	0.5868	0.3824	0.3288	0.3776	0.6979	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=1) $	-1.3968	-1.5370	-1.6319	-1.5754	-1.3136	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
The proposed CIFWBM operator $ (p=1;q=0) $	-1.3485	-1.5108	-1.6119	-1.5700	-1.2525	$ \mathcal{H}_5 \succ \mathcal{H}_1 \succ \mathcal{H}_2 \succ \mathcal{H}_4 \succ \mathcal{H}_3 $
Abbreviations. IFWA: Intuitionistic fuzzy weighted averaging; IFWG: Intuitionistic fuzzy weighted geometric; IFEWA: intuitionistic fuzzy Einstein weighted averaging; IFPWA: intuitionistic fuzzy power weighted averaging; IFWBM: intuitionistic fuzzy weighted Bonferroni mean; IFGIA: intuitionistic fuzzy geometric interactive averaging; IFEWGIA: intuitionistic fuzzy Einstein weighted geometric interactive averaging; IFHWA: intuitionistic fuzzy Hamacher weighted averaging; CIFWBM: complex intuitionistic fuzzy weighted Bonferroni mean.

Table 7. The characteristic comparison of different approaches

Method	Captures interrelationship among arguments	Ability to capture information using complex numbers	Ability to handle two-dimensional information	Ability to integrate Information	Flexible according to decision-maker's preferences
In [37]	$ \times $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $
In [19]	$ \times $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $
In [3]	$ \times $	$ \checkmark $	$ \checkmark $	$ \times $	$ \checkmark $
In [36]	$ \times $	$ \checkmark $	$ \checkmark $	$ \times $	$ \times $
In [42]	$ \checkmark $	$ \times $	$ \times $	$ \checkmark $	$ \checkmark $
In [41]	$ \times $	$ \times $	$ \times $	$ \checkmark $	$ \times $
In [39]	$ \times $	$ \times $	$ \times $	$ \checkmark $	$ \times $
The proposed approach	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \checkmark $

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Method	Captures interrelationship among arguments	Ability to capture information using complex numbers	Ability to handle two-dimensional information	Ability to integrate Information	Flexible according to decision-maker's preferences
In [37]	$ \times $	$ \checkmark $	$ \checkmark $	$ \checkmark $	$ \times $
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2020 Impact Factor: 1.801

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2020 Impact Factor: 1.801

American Institute of Mathematical Sciences

Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

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American Institute of Mathematical Sciences

Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

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